![Children of the Cosmos
Presenting a Toy Model of Science with a Supporting
Cast of Infinitesimals
Sylvia Wenmackers
[A]ll our science, measured against reality, is primitive and
childlike – and yet it is the most precious thing we have.
Albert Einstein [1, p. 404]
[…] I seem to have been only like a boy playing on the
sea-shore, and diverting myself in now and then finding a
smoother pebble or a prettier shell than ordinary, whilst the
great ocean of truth lay all undiscovered before me.
Isaac Newton [2, p. 54]
Abstract Mathematics may seem unreasonably effective in the natural sciences, in
particular in physics. In this essay, I argue that this judgment can be attributed, at least
in part, to selection effects. In support of this central claim, I offer four elements. The
first element is that we are creatures that evolved within this Universe, and that our
pattern finding abilities are selected by this very environment. The second element
is that our mathematics—although not fully constrained by the natural world—is
strongly inspired by our perception of it. Related to this, the third element finds
fault with the usual assessment of the efficiency of mathematics: our focus on the
rare successes leaves us blind to the ubiquitous failures (selection bias). The fourth
element is that the act of applying mathematics provides many more degrees of
freedom than those internal to mathematics. This final element will be illustrated by
the usage of ‘infinitesimals’ in the context of mathematics and that of physics. In
1960, Wigner wrote an article on this topic [4] and many (but not all) later authors
have echoed his assessment that the success of mathematics in physics is a mystery.
The above quote is attributed to Isaac Newton shortly before his death (so in 1727 our shortly
before), from an anecdote in turn attributed to [Andrew Michael] Ramsey by J. Spence [2]. See also
footnote 31 in [3].
S. Wenmackers (B)
KU Leuven, Centre for Logic and Analytic Philosophy, Institute of Philosophy,
Kardinaal Mercierplein 2—Bus 3200, 3000 Leuven, Belgium
e-mail: sylvia.wenmackers@hiw.kuleuven.be
S. Wenmackers
University of Groningen, Faculty of Philosophy, Oude Boteringestraat 52,
9712 GL Groningen, The Netherlands
© Springer International Publishing Switzerland 2016
A. Aguirre et al. (eds.), Trick or Truth?, The Frontiers Collection,
DOI 10.1007/978-3-319-27495-9_2
5](https://image.slidesharecdn.com/4619-250619030227-14ec492c/85/Trick-or-Truth-The-Mysterious-Connection-Between-Physics-and-Mathematics-Anthony-Aguirre-17-320.jpg)
![6 S. Wenmackers
At the end of this essay, I will revisit Wigner and three earlier replies that harmonize
with my own view. I will also explore some of Einstein’s ideas that are connected
to this. But first, I briefly expose my views of science and mathematics, since these
form the canvass of my central claim.
Toy Model of Science
Science can be viewed as a long-lasting and collective attempt at assembling an
enormous jigsaw puzzle. The pieces of the puzzle consist of our experiences (in par-
ticular those that are intersubjectively verifiable) and our argumentations about them
(often in the form of mathematical models and theories). The search for additional
pieces is part of the game. Any piece that we add to the puzzle at one time may be
removed later on. Nobody knows how many pieces there are, what the shape of the
border looks like, or whether the pieces belong to the same puzzle at all. We assume
optimistically that this is the case, indeed, and we attempt to connect all the pieces
of the puzzle that have been placed on the table so far.1
So, like Einstein and (allegedly) Newton in the quotes appearing on the title page,
I view science as a playful and limited activity, which is at the same time a highly
valuable and unprecedented one. Scientific knowledge is fallible, but there is no better
way to obtain knowledge. Hence, it seems wise to base our other epistemic endeavors
(such as philosophy) on science—a position known as ‘naturalism’. In addition,
there is no more secure foundation for scientific knowledge beyond science itself.
The epistemic position of ‘coherentism’ lends support to the positive and optimistic
project of science. It has been phrased most evocatively by Otto Neurath [7, p. 206]:
Like sailors we are, who must rebuild their ship upon the open sea, without ever being able
to put it in a dockyard to dismantle it and to reconstruct it from the best materials.2
We are in the middle of something and we are not granted the luxury of a fresh start.
Hence, we cannot analyze the apparent unreasonable effectiveness of mathematics
in science from any better starting point either. Condemned we are to deciphering
the issue from the incomplete picture that emerges from the scientific puzzle itself,
while its pieces keep moving. A dizzying experience.
Since I mentioned “mathematical models and theories”, I should also express
my view on those.3
To me, mathematics is a long-lasting and collective attempt at
1Making these connections involves developing narratives. Ultimately, science is about storytelling.
“The anthropologists got it wrong when they named our species Homo sapiens (‘wise man’). In any
case it’s an arrogant and bigheaded thing to say, wisdom being one of our least evident features.
In reality, we are Pan narrans, the storytelling chimpanzee.”—Ian Stewart, Jack Cohen, and Terry
Pratchett (2002) [6, p. 32].
2This is my translation of the German quote [7, p. 206]: “Wie Schiffer sind wir, die ihr Schiff auf
offener See umbauen müssen, ohne es jemals in einem Dock zerlegen und aus besten Bestandteilen
neu errichten zu können.”
3In the current context, I differentiate little between ‘models’ and ‘theories’. For a more detailed
account of scientific models, see [5].](https://image.slidesharecdn.com/4619-250619030227-14ec492c/85/Trick-or-Truth-The-Mysterious-Connection-Between-Physics-and-Mathematics-Anthony-Aguirre-18-320.jpg)
![8 S. Wenmackers
is not innate (unfortunately, since otherwise we would not have such a hard time
learning or teaching mathematics), but there are robust findings that very young chil-
dren (as well as newborns of non-human animals, for that matter) possess numerical
abilities [8, 9]. So, we have innate cognitive abilities, that allow us to learn how
to count and—with further effort—to study and to develop more abstract forms of
mathematics.
This raises the further question as to the origin of these abilities. To answer it, we
rely on the coherent picture of science, which tells us this: if our senses and reasoning
did not work at all, at least to an approximation sufficient for survival, our ancestors
would not have survived long enough to raise offspring and we would not have come
into being. Among the traits that have been selected, our ancestors passed on to us
certain cognitive abilities (as well as associated vices: more on this below). On this
view, we owe our innate numerical abilities to the biological evolution of our species
and its predecessors.
Let me give a number of examples to illustrate how our proto-mathematical capac-
ities might have been useful in earlier evolutionary stages of our species. Being able to
estimate and to compare the number of fruits hanging from different trees contributes
to efficient foraging patterns. So does the recognition of regional and seasonal5
pat-
terns in the fruition of plants and the migration of animals. And the ability to plan
future actions (rather than only being able to react to immediate incentives) requires
a crude form of extrapolation of past observations. These traits, which turned out
to be advantageous during evolution, lie at the basis of our current power to think
abstractly and to act with foresight.
Our current abilities are advanced, yet limited. Let us first assess our extrapolative
capacities: we are far from perfect predictors of the future. Sometimes, we fail to
take into account factors that are relevant, or we are faced with deterministic, yet
intrinsically chaotic systems. Consider, for example, a solar eclipse. An impending
occultation is predicted many years ahead. However, whether the weather will be
such that we can view the phenomenon from a particular position on the Earth’s
surface, that is something we cannot predict reliably a week ahead. Let us then
turn to the more basic cognitive faculty of recognizing patterns. We are prone to
patternicity, which is a bias that makes us see patterns in accidental correlations [10].
This patternicity also explains why we like to play ‘connect the dots’ while looking
at the night sky: our brains are wired to see patterns in the stars, even though the
objects we thus group into constellations are typically not in each other’s vicinity;
the patterns are merely apparent from our earthbound position.
In our evolutionary past, appropriately identifying many patterns yielded a larger
advantage than the disadvantage due to false positives. In the case of a tiger, it is clear
that one false negative can be lethal. But increasing appropriate positives invariable
comes at the cost of increasing false positives as well.6
As a species, we must make do without venom or an exoskeleton, alas, but we have
higher cognitive abilities that allow us to plan our actions and to devise mathematics.
5Or ‘spatiotemporal’, if you like to talk like a physicist.
6The same trade-off occurs, for instance, in medical testing and law cases.](https://image.slidesharecdn.com/4619-250619030227-14ec492c/85/Trick-or-Truth-The-Mysterious-Connection-Between-Physics-and-Mathematics-Anthony-Aguirre-20-320.jpg)
![10 S. Wenmackers
variation produces many unviable results. Evolution is squandermanious—quite the
opposite of efficient. The selection process is mainly driven by cultural factors, which
are internal to mathematics (favoring theories that exhibit epistemic virtues such as
beauty and simplicity). But, as we saw in the previous paragraph, empirical factors
come into play as well, mediated by external interactions with science. Although
mathematics is often described as an a priori activity, unstained by any empirical
input, this description itself involves an idealization. In reality, there is no a priori.
Mathematics Fails Science More Often Than Not
For each abstraction, many variations are possible, the majority of which are not
applicable to our world in any way. The effectiveness perceived by Wigner [4] may
be due to yet another form of selection bias: one that makes us prone to focus on the
winners, not the bad shots. Moreover, even scientific applications of mathematics that
are widely considered to be highly successful have a limited range of applicability
and even within that range they have a limited accuracy.
Among the mathematics books in university libraries, many are filled with theories
for which not a single real world application has been found.8
We could measure
the efficiency of mathematics for the natural science as follows: divide the number
of pages that contain scientifically applicable results by the total number of pages
produced in pure mathematics. My conjecture is that, on this definition, the efficiency
is very low. In the previous section we saw that research, even in pure mathematics,
is biased towards the themes of the natural sciences. If we take this into account, the
effectiveness of mathematics in the natural sciences does come out as unreasonable—
unreasonably low, that is.9
Maybe it was unfair to focus on pure mathematics in the proposed definition
for efficiency? A large part of the current mathematical corpus deals with applied
mathematics,fromdifferentialequationstobio-statistics.Ifwemeasuretheefficiency
by dividing the number of ‘applicable pages’ by the total number of pages produced
in all branches of mathematics, we certainly get a much higher percentage. But, now,
the effectiveness of mathematics in the natural sciences appears reasonable enough,
sinceresearchandpublicationsinappliedmathematicsare(rightfully)biasedtowards
real world applicability.
At this point, you may object that Wigner made a categorical point that there is
some part of mathematics at all that works well, even if this does not constitute all
or most of mathematics. I am sympathetic to this objection (and the current point is
the least important one in my argument), but then what is the contrasting case: that
8This is fine, of course, since this is not the goal of mathematics.
9Here, I recommend humming a Shania Twain song: “So, you’re a rocket scientist. That don’t
impress me much.” If you are too young to know this song, consult your inner teenager for the
appropriate dose of underwhelmedness.](https://image.slidesharecdn.com/4619-250619030227-14ec492c/85/Trick-or-Truth-The-Mysterious-Connection-Between-Physics-and-Mathematics-Anthony-Aguirre-22-320.jpg)
![Children of the Cosmos 11
no mathematics would describe anything in the Universe? I offer some speculations
about this in section “A Speculative Question Concerning the Unthinkable”.
Abundant Degrees of Freedom in Applying Mathematics:
The Case of Infinitesimals
I once attended a lecture in which the speaker claimed that “There is a matter of
fact about how many people are in this room”. Unbeknownst to anyone else in that
room, I was pregnant at the time, and I was unsure whether an unborn child should be
included in the number of people or not. To me, examples like this show that we can
apply mathematically crisp concepts (such as the counting numbers) to the world,
but only because other concepts (like person or atom) are sufficiently vague.
The natural sciences aim to formulate their theories in a mathematically precise
way, so it seems fitting to call them the ‘exact sciences’. However, the natural sci-
ences also allow—and often require—deviations from full mathematical rigor. Many
practices that are acceptable to physicists—such as order of magnitude calculations,
estimations of errors, and loose talk involving infinitesimals—are frowned upon by
mathematicians. Moreover, all our empirical methods have a limited range and sen-
sitivity, so all experiments give rise to measurement errors. Viewed as such, one may
deny that any empirical science can be fully exact. In particular, systematic discrep-
ancies between our models and the actual world can remain hidden for a long time,
provided that the effects are sufficiently small, compared to our current background
theories and empirical techniques.
Einstein put it like this: “As far as the laws of mathematics refer to reality, they
are not certain; and as far as they are certain, they do not refer to reality” [11, p. 28].
To illustrate this point, I will concentrate on the calculus—the mathematics of differ-
ential and integral equations—and consider the role of infinitesimals in mathematics
as well as in physics.
In mathematics, infinitesimals played an important role during the development of
the calculus, especially in the work of Leibniz [12], but also in that of Newton (where
they figure as ‘evanescent increments’) [13]. The development of the infinitesimal
calculus was motivated by physics: geometric problems in the context of optics, as
well as dynamical problems involving rates of change. Berkeley [14] ridiculed these
analysts as employing “ghosts of departed quantities”. It has taken a long time to
find a consistent definition of infinitesimals that holds up to the current standards of
mathematical rigour, but meanwhile this has been achieved [15]. The contemporary
definition of infinitesimals considers them in the context of an incomplete, ordered
field of ‘hyperreal’ numbers, which is non-Archimedean: unlike the field of real
numbers, it does contain non-zero, yet infinitely small numbers (infinitesimals).10
The alternative calculus based on hyperreal numbers, called ‘non-standard analysis’
10Here, I mean by infinitesimals numbers larger than zero, yet smaller than 1/n for any natural
number n.](https://image.slidesharecdn.com/4619-250619030227-14ec492c/85/Trick-or-Truth-The-Mysterious-Connection-Between-Physics-and-Mathematics-Anthony-Aguirre-23-320.jpg)
![12 S. Wenmackers
(NSA), is conceptually closer to Leibniz’s original work (as compared to standard
analysis).
While infinitesimals have long been banned from mathematics, they remained in
fashion within the sciences, in particular in physics: not only in informal discourse,
but also in didactics, explanations, and qualitative reasoning. It has been suggested
that NSA can provide a post hoc justification for how infinitesimals are used in
physics [16]. Indeed, NSA seems a very appealing framework for theoretical physics:
it respects how physicists are already thinking of derivatives, differential equations,
series expansions, and the like, and it is fully rigorous.11
Rephrasing old results in the language of NSA may yield new insights. For
instance, NSA can be employed to make sense of classical limits in physics: classi-
cal mechanics can be modelled as quantum mechanics with an infinitesimal Planck
constant [22]. Likewise, Newtonian mechanics can be modelled as a relativity theory
with an infinite maximal speed, c (or infinitesimal 1/c).
Infinitesimal numbers are indistinguishable from zero (within the real numbers),
yet distinct from zero (as can be made explicit in the hyperreal numbers). This
is suggestive of a physical interpretation of infinitesimals as ‘currently unobserv-
able quantities’. The ontological status of unobservables is an important issue in the
realism–anti-realism debate [23]. Whereas constructive empiricists interpret ‘observ-
ability’ as ‘detectability by the human, unaided senses’ [24], realists regard ‘observ-
ability’ as a vague, context-dependent notion [25]. When an apparatus with better
resolving power is developed, some quantities that used to be unobservably small
become observable [26, 27]. This shift in the observable-unobservable distinction
can be modelled by a form of NSA, called relative analysis, as a move to a finer
context level [28]. Doing so requires the existing static theory to be extended by new
principles that constrain the allowable dynamics [29].
The interpretation of (relative) infinitesimals as (currently) unobservable quanti-
ties is suggestive of why the calculus is so applicable to the natural sciences: it appears
that infinitesimals provide scientists with the flexibility they need to fit mathematical
theories to the empirically accessible world. To return to the jigsaw puzzle analogy of
section “Toy Model of Science”: we need some tolerance at the edges of the pieces.
If the fit is too tight, it becomes impossible to connect them at all.
A Speculative Question Concerning the Unthinkable
Could our cosmos have been different—so different that a mathematical description
of it would have been fundamentally impossible (irrespective of whether life could
11It hasbeenshownthat physical problemscanbe rephrasedintermsofNSA[17],bothinthe context
of classical physics (Lagrangian mechanics [18]) and of quantum mechanics (quantum field theory
[19], spin models [20], relativistic quantum mechanics [21], and scattering [18]). Apart from formal
aspects (mathematical rigour), such a translation also offers more substantial advantages, such as
easier (shorter) proofs.](https://image.slidesharecdn.com/4619-250619030227-14ec492c/85/Trick-or-Truth-The-Mysterious-Connection-Between-Physics-and-Mathematics-Anthony-Aguirre-24-320.jpg)
![Children of the Cosmos 13
emerge in it)? Some readers may have the impression that I have merely explored
issues in the vicinity of this mystery, without addressing it directly.
Before I indulge in this speculation, it may be worthwhile to remember that the
very notion of a ‘cosmos’ emerged in ancient Greek philosophy, with the school of
Pythagoras, where it referred to the order of the Universe (not the Universe itself).
It is closely related to the search for archai or fundamental ordering principles. It
is well known that the Pythagorians took the whole numbers and—by extension—
mathematics as the ordering principle of the Universe. Their speculations about a
mathematically harmonious music of the spheres resonated with Plato and Johannes
Kepler (the great astronomer, but also the last great neoplatonist). Since these archai
had to be understandable to humans, without divine intervention or mystical revela-
tion, they had to be limited in number and sufficiently simple. So, the idea that the
laws of nature have to be such that they can be printed on the front of a T-shirt, goes
back to long before the invention of the T-shirt.12
In this sense, the answer to the
speculative question at the start of this section is ‘no’ and trivially so, for otherwise
it would not be a cosmos. Yet, even if we understand ‘our cosmos’ as ‘the Universe’,
there is a strong cultural bias to answer the speculative question in the negative.
In section “A Natural History of Mathematicians”, I considered our proto-
mathematical abilities as well as their limits. At least in some areas, our predic-
tions do better than mere guesses. This strongly suggests that there are patterns in
the world itself—maybe not the patterns that we ascribe to it, since these may fail, but
patterns all the same. It is then often taken to be self-evident that these patterns must
be mathematical, but to me this is a substantial additional assumption. On my view
of mathematics, the further step amounts to claiming that nature itself is—at least
in principle—understandable by humans. I think that all we understand about nature
are our mathematical representations of it.13
Ultimately, reality is not something to
be understood, merely to be. (And, for us, to be part of.)
When we try to imagine a world that would defy our mathematical prowess, it is
temptingtothinkofaworldthatistotallyrandom.However,thisattemptisfutile.Pure
randomness is a human idealization of maximally unpredictable outcomes (like a
perfectly fair lottery [30]). Yet, random processes are very well-behaved: they consist
of events that may be maximally unpredictable in isolation, but collectively they
produce strong regularities. It is no longer a mystery to us how order emerges from
chaos. In fact, we have entire fields of mathematics for that, called probability theory
and statistics, which are closely related to branches of physics, such as statistical
mechanics.
As a second attempt, we could propose a Daliesque world, in which elements
combine in unprecedented ways and the logic seems to change midgame: rigid clocks
become fluid, elephants get stilts, and tigers emerge from the mouths of fish shooting
12In case this remark made you wonder: the T-shirt was invented about a century ago.
13My view of mathematics might raise the question: “Why, then, should we expect that anything as
human and abstract as mathematics applies to concrete reality?” I think this question is based on a
false assumption, due to prolonged exposure to Platonism—remnants of which are abundant in our
culture.](https://image.slidesharecdn.com/4619-250619030227-14ec492c/85/Trick-or-Truth-The-Mysterious-Connection-Between-Physics-and-Mathematics-Anthony-Aguirre-25-320.jpg)
![14 S. Wenmackers
from a pomegranate. Yet, even such surrealistic tableaus have meta-regularities of
their own. Many people are able to recognize a Dalí painting instantly as his work,
even if they have not seen this particular painting before. Since we started from
human works of art, unsurprisingly, the strategy fails to outpace our own constrained
imagination.
At best, I can imagine a world in which processes cannot be summarized or
approximated in a meaningful way. Our form of intelligence is aimed at finding the
gist in information streams, so it would not help us in this world (in which it would
not arise spontaneously by biological evolution either). In any case, what I can
imagine about such a world remains very vague—insufficient for any mathematical
description. Maybe there are better proposals out there?
Max Tegmark has put forward an evocative picture of the ultimate multiverse as
consisting of all the orderings that are mathematically possible [31]. (See also Marc
Séguin’s contribution [32].) Surely, this constitutes a luscious multiplicity. From my
view of mathematics as constrained imagination, however, the idea of a mathematical
multiverse is still restricted by what is thinkable by us, humans. Aristotle described
us as thinking animals, but for the current purpose ‘mathematizing mammals’ may
fit even better. My diagnosis of the situation is that the speculative question asks us
to boldly go even beyond Tegmark’s multiverse and thus to exceed the limits of our
cognitive kung fu: even with mathematics, we cannot think the unthinkable.
Reflections on Wigner and Einstein
In this section, I return to Wigner [4] and compare my own reflections to earlier
replies by Hamming [33], Grattan-Guinness [34], and Abbott [35]. I end with a
short reflection on some ideas of Einstein [11] that predate Wigner’s article by four
decades.
Wigner’s Two Miracles
Wigner wrote about “two miracles”: “the existence of the laws of nature” and “the
human mind’s capacity to divine them” [4, p. 7]. First and foremost, I hope that my
essay helps to see that we need not presuppose the former to understand the latter:
it is by assuming that the Universe forms a cosmos that we have started reading
laws into it. Galileo Galilei later told us that those laws are mathematical ones. The
very term “laws of nature” may be misleading and for this reason, I avoided it so far
(except for the remark of fitting them on a T-shirt). The fact that our so-called laws
can be expressed with the help of mathematics should be telling, since that is our
science of patterns. When we open Galileo’s proverbial book of nature, we find it
filled with our own handwriting.](https://image.slidesharecdn.com/4619-250619030227-14ec492c/85/Trick-or-Truth-The-Mysterious-Connection-Between-Physics-and-Mathematics-Anthony-Aguirre-26-320.jpg)
![Children of the Cosmos 15
To illustrate the unreasonable effectiveness of mathematics, Wigner offered the
following analogy for it: consider a man with many keys in front of many doors, who
“always hits on the right key on the first or second try” [4, p. 2]. Lucky streaks like
this may seem to require further explanation. However, if there are many people, each
with many keys, it becomes likely that at least one of them will have an experience
like Wigner’s man—and no further explanation is needed (see also Hand [36]). There
are indeed many people active in mathematics and science, and few of them succeed
“on the first or second try”—or at all. In the essay, I argued that the perceived
effectiveness of mathematics in physics can be diagnosed in terms of selection bias.
The same applies to the metaphor Wigner presented.
Wigner found it hard to believe that the perfection of our reasoning power was
brought about by Darwin’s process of natural selection [4, p. 3]. Ironically, the
selection bias that he may have fallen prey to is a good illustration of the lack of
perfection of our reasoning powers. To be clear, I do not claim that Darwin’s theory
ofbiologicalevolutionsufficestoexplainthesuccessof(certainpartsof)mathematics
in physics. However, a similar combination of variation and selection is at work in the
evolution of mathematics and science. See also Pólya, as cited by Grattan-Guinness,
who has given an iterative or evolutionary description of the development of science
[37, Vol. 2, p. 158].
Previous Replies to Wigner
In his description of mathematics, Wigner wrote about “defining concepts beyond
those contained in the axioms” [4, p. 3]. Wigner did not, however, reflect on where
those axioms come from in the first place. This has been criticized by Hamming [33],
a mathematician who worked on the Manhattan Project and for Bell Labs. Axioms
or postulates are not specified upfront; instead, mathematicians may try various pos-
tulates until theorems follow that harmonize with their initial vague ideas. Hamming
cited the Pythagorean theorem and Cauchy’s theorem as examples: if mathemati-
cians would have started out with a system in which those crucial results would not
hold, then—according to Hamming—they would have changed their postulates until
they did. And, of course, the initial vague ideas are thoughts produced by beings
entrenched in the physical world. This brings us to Putnam, who pointed out that
mathematical knowledge resembles empirical knowledge in many respects: “the cri-
terion of success in mathematics is the success of its ideas in practice” [38, p. 529].
Wigner did concede that not any mathematical concept will do for the formulation
of laws of nature in physics [4, p. 7], but he claimed that “in many if not most cases”
these concepts were developed “independently by the physicist and recognized then
as having been conceived before by the mathematician” [4, p. 7] (my emphasis). I
think this part is misleading: there is a lot of interaction between mathematics and
physics and what is being ‘recognized’ is actually the finding of a new analogy. We
may think that we are merely discovering a similarity, when we are really forging
new connections, which may subtly alter both sides. This is a creative element of](https://image.slidesharecdn.com/4619-250619030227-14ec492c/85/Trick-or-Truth-The-Mysterious-Connection-Between-Physics-and-Mathematics-Anthony-Aguirre-27-320.jpg)
![16 S. Wenmackers
great importance within mathematics as well as in finding applications to other fields,
in which our patternicity may be a virtue rather than a vice.
Both aspects have been illuminated by Grattan-Guinness, a historian of mathe-
matics, who argued that “much mathematics has been motivated by interpretations in
the sciences” [34, p. 7]. He stressed the importance of analogies within mathematics
and between mathematics and natural science and he gave historical examples in
which mathematics and physics take turns in reshaping earlier concepts. Moreover,
he remarked that there are many analogies that can be tried (somewhat similar to the
ideas in section “Abundant Degrees of Freedom in Applying Mathematics: The Case
of Infinitesimals”), but that only the successful ones are taken into account when
assessing the effectiveness of mathematics (postselection as in section “Mathematics
as Constrained Imagination”).
My essay mainly focused on elementary mathematics and simple models. Of
course, there are very complicated mathematical theories in use in advanced physics.
In relation to this, Grattan-Guinness observed that “by around 1900 linearisation had
become something of a fixation” [34, p. 11], but he also discussed the subsequent
“desimplification” or “putting back in the theory effects and factors that had been
deliberately left out” [34, p. 13].
In light of my discussion of infinitesimals in this essay, it is curious to observe
that Grattan-Guinness and Hamming referred to them too. Grattan-Guinness spoke
approvingly of the Leibniz-Euler approach to the calculus because it “often has a
better analogy content to the scientific context” [34, p. 15]. Hamming even mentioned
NSA, but only as an example of the observation that “logicians can make postulates
that put still further entities on the real line” [33, p. 85].
Recently, the reply by Hamming has been developed further by Abbott, a professor
in electrical engineering [35]. Whereas Hamming described his recurrent feeling that
“God made the universe out of complex numbers” [33, p. 85], Abbott described the
complex numbers as “simply a convenience for describing rotations”, concluding
that “the ubiquity of complex numbers is not magical at all” [35, p. 2148].
More specifically, Abbott adds two points to Hamming’s earlier observations.
Abbott’s first addition is that “all physical laws and mathematical expressions of
those laws are […] necessarily compressed due to the limitations of the human mind”
[35, p. 2150]. He explains that the associated loss of information does not preclude
usefulness “provided the effects we have neglected are small”, which lends itself
perfectly to a rephrasing in terms of ‘my’ relative infinitesimals. Abbott’s second
addition is that “the class of successful mathematical models is preselected”, which
he described as a “Darwinian selection process” [35, p. 2150]. Like I did here, Abbott
warned his readers not to overstate the effectiveness of mathematics. Moreover, as
an engineer, he is well aware that “when analytical methods become too complex,
we simply resort to empirical models and simulations” [35, p. 2148].
The title of Abbot’s piece, “The reasonable ineffectiveness of mathematics”, and
the general anti-Platonist stance agree with the views exposed in the current essay.
In addition, Abbott tried to show that this debate is relevant, even for those who
prefer to “shut up and calculate”, because “there is greater freedom of thought,
once we realize that mathematics is something we entirely invent as we go along”
[35, p. 2152].](https://image.slidesharecdn.com/4619-250619030227-14ec492c/85/Trick-or-Truth-The-Mysterious-Connection-Between-Physics-and-Mathematics-Anthony-Aguirre-28-320.jpg)
![Children of the Cosmos 17
Einstein’s Philosophy of Science
In 1921, so almost forty years before Wigner wrote his article, Einstein gave an
address to the Prussian Academy of Sciences in Berlin titled “Geometrie und
Erfahrung”. The expanded and translated version contains the following passage
(which includes the sentence already quoted in section “Abundant Degrees of
Freedom in Applying Mathematics: The Case of Infinitesimals”):
At this point an enigma presents itself which in all ages has agitated inquiring minds. How
can it be that mathematics, being after all a product of human thought which is independent
of experience, is so admirably appropriate to the objects of reality? Is human reason, then,
without experience, merely by taking thought, able to fathom the properties of real things.
In my opinion the answer to this question is, briefly, this: As far as the laws of mathematics
refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
[11, p. 28]
On the one hand, we read of “an enigma”, which seems to set the stage for Wigner’s
later question. On the other hand, we are not dealing with a Platonist conception of
mathematics here, since Einstein describes it as “a product of human thought”. Yet,
how are we to understand Einstein’s addition that mathematics is “independent of
experience”? This becomes clearer in the remainder of his text: geometry stems from
empirical “earth-measuring”, but modern axiomatic geometry, which allows us to
consider multiple axiomatisations (including non-Euclidean ones), remains silent on
whether any of these axiom schemes applies to reality [11, pp. 28–32]. Einstein refers
approvingly to Schlick’s view that axioms in the modern sense function as implicit
definitions, which is in agreement with Hilbert’s formalist position.14
Regarding his own answer to the question, Einstein further explains that axiomatic
geometry can be supplemented with a proposition to relate mathematical concepts
to objects of experience: “Geometry thus completed is evidently a natural science”
[11, p. 32]. My essay agrees with Einstein’s answer: the application of mathematics
supplies additional degrees of freedom external to mathematics and we can never be
sure that the match is perfect, since empirical precision is always limited. On other
occasions, Einstein also pointed out that Kant’s a priori would better be understood
as ‘conventional’: a position close to that of Pierre Duhem (see for instance [39]).
In my late teens, I read a Dutch translation of essays by Einstein. The oldest essay
in that collection stemmed from 1936 [40], so the text from which I quoted does not
appear in that book. However, it is clear that similar ideas have influenced me in my
formative years. It was a pleasure to reexamine some of them here. They contributed
to my decision to become a physicist and a philosopher of physics, which in turn
helped me to write this essay. Hence, the appearance of a text in this collection that
is in agreement with some of Einstein’s view on science may be a selection effect
as well.
14For the influence of Moritz Schlick on Einstein’s ideas, see [39].](https://image.slidesharecdn.com/4619-250619030227-14ec492c/85/Trick-or-Truth-The-Mysterious-Connection-Between-Physics-and-Mathematics-Anthony-Aguirre-29-320.jpg)
![305
It must be sufficient for us that Ignatius Loyola had now gotten the
mastery of Francis Xavier so perfectly that he could be “applied to
the Spiritual Exercises, the furnace in which he [Loyola] was
accustomed to refine and purify his chosen vessels.” A sister
of the future missionary had become one of the Barefooted Clares,
and had aided in dissuading her father from interference. And now
we behold Xavier praying with hands and feet tightly bound by
cords; or journeying with similar cords about his arms and the calves
of his legs until inflammation and ulceration ensued. There were
now nine of these converts, but this man outdid the others in his
austerities, and finally travelled on foot with them to meet Loyola at
Venice in 1537. The society had really been formed on August 15th,
1534, at Montmartre near Paris, and this was but its natural outward
movement.
At Venice, on January 8th, 1537, they again met their leader and
were assigned for duty to the two hospitals of the city. That of the
“Incurables” fell to Xavier’s share, and we read that with the morbid
devotion characteristic of a devout student of the Exercises, he
determined now to conquer his natural repugnance to disease. In
the course of his duties he had an unusually hideous ulcer to dress
for one of the patients. And the authentic history relates that
“encouraging himself to the utmost, he stooped down, kissed the
pestilent cancer, licked it several times with his tongue, and finally
sucked out the virulent matter to the last drop.” (Bartoli and Maffei,
p. 35.) There could be nothing worse than that certainly. And a man
who had resolutely sounded this deepest abyss of self-abandonment
was marked for the highest honor that the new society could
bestow. We cannot doubt Xavier’s sincerity, but the gigantic horror of
this performance is of a sort to place the man who has achieved it
upon an eminence apart from less daring minds. It was Loyola’s way
of facing human nature and forcing it to concede the supreme self-
devotion of his followers. The world looks with amazement upon
such actions, but when it sees them, it yields a kind of stupefied
allegiance to those who have thus rushed beyond the bounds. And
to a close analysis there is as much concealed spiritual pride about](https://image.slidesharecdn.com/4619-250619030227-14ec492c/85/Trick-or-Truth-The-Mysterious-Connection-Between-Physics-and-Mathematics-Anthony-Aguirre-35-320.jpg)
![307
after such exercises and an additional repose upon a bed of thorns,
was “accustomed to converse with God.” [Aliquando inter spinas
volutaret sic Deum alloqui solita.] This topic, with its allied
suggestions, is altogether out of our present scope; but in order to
see Xavier as he was, we must appreciate to what extent his spirit
was subdued before his belief.
This was the man, tested and edged and tempered, to whom was
now committed the “salvation of the Indies.” It was during the
papacy of Paul III., the same Pope who excommunicated
Henry VIII. of England. And Xavier, who had practised many
austerities both in life and in behavior, was at first sent to Bologna,
while Loyola, with Faber and Laynez, went to Rome. It was
subsequently at Rome that Xavier had his famous vision, in which he
awoke crying, “Yet more, O Lord, yet more!” for he fancied that—as
the Apostle Paul once did—he had beheld his future career and was
glorying in trials and persecutions. Especially did he often have a
dream in which he seemed to be carrying an Indian on his shoulders
and toiling with him over the roughest and hardest roads. And when
at last Govea, the Rector of the College of Ste. Barbe, happened to
be in Rome, Ignatius and his companions were brought by him to
the notice of John III. of Portugal, and the king desired to have six
of them for use in India. The Pope did not show any special desire to
secure their services, and when the question came up he referred it
to Ignatius to decide it as he pleased. That sagacious general
objected to taking six from ten and leaving only four to the rest of
the world, for his ambition now extended to the orb of the earth. He
accordingly chose Rodriguez and Bobadilla for India, men who were
evidently well selected, for the first became a great propagandist in
Portugal, and the other was a decided obstacle to the Reformation in
Germany. When Rodriguez, however, fell ill with an intermittent fever
Xavier naturally occurred to Loyola as the proper substitute. He
therefore commissioned him for the service, and the worn and
wasted ascetic patched up his old coat, said farewell to his friends,
and having craved the Pope’s blessing, set off from Rome with the
Portuguese Ambassador, Mascarenhas, on March 16th, 1540. He](https://image.slidesharecdn.com/4619-250619030227-14ec492c/85/Trick-or-Truth-The-Mysterious-Connection-Between-Physics-and-Mathematics-Anthony-Aguirre-37-320.jpg)
![318
Canticles (Cantica); and many outside it will not fit into any such
definition of what a hymn is. But it illustrates the general character
and purpose of the hymns of the Roman and other breviaries, as
designed for a special temporal or personal application by way of
supplement to the Psalter.
At present the Roman Breviary, prepared with the sanction of the
Council of Trent, has driven nearly all the others out of use. But at
the era of the Reformation there was a great number of
breviaries, every diocese and religious order having a right to
its own. Panzer enumerates no less than seventy-one which were
printed before 1536, some of them in several editions.
[18]
Even now
the Roman Breviary is supplemented by special services in honor of
the saints of each order or country, and by services of a more
general kind which are peculiar to some localities. But in Luther’s
time the endless variety in breviaries and missals formed a striking
feature of the confusion which to his mind characterized the Church
of Rome.
With the development of a more fastidious taste, through the study
of the Latin classics as literary models, there arose in the sixteenth
century, and even before the Reformation, a demand for a
reformation of the Breviary. Besides its defects of form, such as
violations of Latin grammar, the constant use of terms which grated
on the ears of the humanists, and the use of hymns in which rhyme
rather added to the offence of want of correct metre, the contents of
the Breviary were found faulty by a critical age. The selections from
the Fathers to be read by way of homily were in some cases from
spurious works; and the narratives of saints’ lives for the days
dedicated to them were not always edifying, and in some cases
palpably untrue. It became a proverbial saying that a person lied like
the second nocturn office of the Breviary, that being the service in
which these legends are found. But the badness of the Latin and the
metrical faults of the hymns counted for quite as much with the
critics of that day. We hear of a cardinal warning a young cleric not](https://image.slidesharecdn.com/4619-250619030227-14ec492c/85/Trick-or-Truth-The-Mysterious-Connection-Between-Physics-and-Mathematics-Anthony-Aguirre-50-320.jpg)
![319
to be too constant in reading his Breviary, if he wished to preserve
his ear for correct Latinity.
As might have been expected, it was the elegant Medicean Pope Leo
X. who first put his hand to the work of reform. He selected for this
purpose Zacharia Ferreri, Bishop of Guarda-Alfieri, a man of fine
Latin scholarship and some ability as a poet. By 1525 Ferreri had the
hymns for a new Breviary ready, and published them with the
promise of the Breviary itself on the title-page.
[19]
Clement
VII., also of the house of Medici, was Pope when the book
appeared, and he authorized the substitution of these new hymns
for the old, but did not command this.
The book is furnished with an introduction by Marino Becichemi, a
forgotten humanist, who was then professor of eloquence at Padua.
It is worth quoting as exhibiting the attitude of the Renaissance to
the earlier Christian literature. He praises Ferreri as a shining light in
every kind of science, human and divine, prosaic and poetical. He
cannot say too much of the beauty of his style, its gravity and
dignity, its purity, its spontaneity and freedom from artificiality. “That
his hymns and odes, beyond all doubt, will secure him immortality, I
need not conceal. Certainly I have read nothing in Christian poets
sweeter, purer, terser, or brighter. How brief and how copious, each
in its place—how polished! Everywhere the stream flows in full
channel with that antique Roman mode of speech, except where of
full purpose it turns in another direction.” That means how
Ciceronian Ferreri’s speech, except where he remembers that he is a
Christian poet and bishop writing for Christian worshippers. “More
than once have I exhorted him that it belonged to the duty and
dignity of his episcopal (pontificii) office to make public these Church
hymns.”
“You know, my reader, what hymns they sing everywhere in the
temples, that they are almost all faulty, silly, full of barbarism, and
composed without reference to the number of feet or the quantity of](https://image.slidesharecdn.com/4619-250619030227-14ec492c/85/Trick-or-Truth-The-Mysterious-Connection-Between-Physics-and-Mathematics-Anthony-Aguirre-51-320.jpg)
![321
was to use the space thus obtained to insert ampler Scripture
lessons and more Psalms, so that, as in earlier times, the Bible might
be read through once a year and the Psalter once a week. It is this
last feature which has elicited the praise of Protestant liturgists, and
it is known that the Breviary of Quiñonez furnished the basis for the
services of the Anglican Book of Common Prayer, excepting, of
course, the Communion Service. But unfortunately hymnologists are
not able to join in this praise. To get the Psalms said or sung through
once a week, he dealt nearly as ruthlessly with the hymns as if he
were a Seceder.
His Breviary appeared in 1535,
[20]
and for thirty-three years its use
was permitted to ecclesiastics in their private recitation of the
hours. It appeared in a large number of editions in different
parts of Europe, so that its use must have been extensive. But it did
not pass unchallenged. The doctors of the Sorbonne at Paris hurried
into the arena with their condemnation of it before the ink was fully
dry on the first copies. They declared it a thing unheard of to
introduce into Church use a book which was the production of a
single author, and he—as they wrongly alleged—not even a member
of any religious order. Furthermore, he had so shortened and
eviscerated the legends for the saints’ days, besides omitting many,
that nobody could tell what virtues and what miracles entitled them
to commemoration. Above all he had omitted Peter Damiani’s Little
Office of the Blessed Virgin! Much better founded was the objection
to the omission of parts long established in use, such as the
antiphons and many of the hymns. Here we must side with the
Sorbonne against Quiñonez.
It was not until 1568 that the present Roman Breviary appeared.
When the Council of Trent met in its final session in 1562, the first
drafts of a reformed Breviary and Missal were transmitted to the
Fathers by Pius IV.; but they were too busy with questions of
discipline to do more than return these with their approbation. The
work was published by Pius V. in July, 1568, and its use was made](https://image.slidesharecdn.com/4619-250619030227-14ec492c/85/Trick-or-Truth-The-Mysterious-Connection-Between-Physics-and-Mathematics-Anthony-Aguirre-53-320.jpg)
Trick or Truth The Mysterious Connection Between Physics and Mathematics Anthony Aguirre
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- 4. 123 Anthony Aguirre Brendan Foster Zeeya Merali (Eds.) Trick or Truth? The Mysterious Connection Between Physics and Mathematics T H E F R O N T I E R S C O L L E C T I O N
- 5. THE FRONTIERS COLLECTION Series editors Avshalom C. Elitzur Iyar The Israel Institute for Advanced Research, Rehovot, Israel e-mail: avshalom.elitzur@weizmann.ac.il Laura Mersini-Houghton Department of Physics, University of North Carolina, Chapel Hill, NC 27599-3255, USA e-mail: mersini@physics.unc.edu T. Padmanabhan Inter University Centre for Astronomy and Astrophysics (IUCAA), Pune, India Maximilian Schlosshauer Department of Physics, University of Portland, Portland, OR 97203, USA e-mail: schlossh@up.edu Mark P. Silverman Department of Physics, Trinity College, Hartford, CT 06106, USA e-mail: mark.silverman@trincoll.edu Jack A. Tuszynski Department of Physics, University of Alberta, Edmonton, AB T6G 1Z2, Canada e-mail: jtus@phys.ualberta.ca Rüdiger Vaas Center for Philosophy and Foundations of Science, University of Giessen, 35394, Giessen, Germany e-mail: ruediger.vaas@t-online.de
- 6. THE FRONTIERS COLLECTION Series Editors A.C. Elitzur L. Mersini-Houghton T. Padmanabhan M. Schlosshauer M.P. Silverman J.A. Tuszynski R. Vaas The books in this collection are devoted to challenging and open problems at the forefront of modern science, including related philosophical debates. In contrast to typical research monographs, however, they strive to present their topics in a manner accessible also to scientifically literate non-specialists wishing to gain insight into the deeper implications and fascinating questions involved. Taken as a whole, the series reflects the need for a fundamental and interdisciplinary approach to modern science. Furthermore, it is intended to encourage active scientists in all areas to ponder over important and perhaps controversial issues beyond their own speciality. Extending from quantum physics and relativity to entropy, conscious- ness and complex systems—the Frontiers Collection will inspire readers to push back the frontiers of their own knowledge. More information about this series at http://www.springer.com/series/5342 For a full list of published titles, please see back of book or springer.com/series/5342
- 7. Anthony Aguirre • Brendan Foster Zeeya Merali Editors Trick or Truth? The Mysterious Connection Between Physics and Mathematics 123
- 8. Editors Anthony Aguirre Department of Physics University of California Santa Cruz, CA USA Brendan Foster Foundational Questions Institute Decatur, GA USA Zeeya Merali Foundational Questions Institute Decatur, GA USA ISSN 1612-3018 ISSN 2197-6619 (electronic) THE FRONTIERS COLLECTION ISBN 978-3-319-27494-2 ISBN 978-3-319-27495-9 (eBook) DOI 10.1007/978-3-319-27495-9 Library of Congress Control Number: 2015958338 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by SpringerNature The registered company is Springer International Publishing AG Switzerland
- 9. Preface This book is a collaborative project between Springer and The Foundational Questions Institute (FQXi). In keeping with both the tradition of Springer’s Frontiers Collection and the mission of FQXi, it provides stimulating insights into a frontier area of science, while remaining accessible enough to benefit a non-specialist audience. FQXi is an independent, nonprofit organization that was founded in 2006. It aims to catalyze, support, and disseminate research on questions at the foundations of physics and cosmology. The central aim of FQXi is to fund and inspire research and innovation that is integral to a deep understanding of reality, but which may not be readily supported by conventional funding sources. Historically, physics and cosmology have offered a scientific framework for comprehending the core of reality. Many giants of modern science—such as Einstein, Bohr, Schrödinger, and Heisenberg—were also passionately concerned with, and inspired by, deep philosophical nuances of the novel notions of reality they were exploring. Yet, such questions are often over- looked by traditional funding agencies. Often, grant-making and research organizations institutionalize a pragmatic approach, primarily funding incremental investigations that use known methods and familiar conceptual frameworks, rather than the uncertain and often interdisci- plinary methods required to develop and comprehend prospective revolutions in physics and cosmology. As a result, even eminent scientists can struggle to secure funding for some of the questions they find most engaging, while younger thinkers find little support, freedom, or career possibilities unless they hew to such strictures. FQXi views foundational questions not as pointless speculation or misguided effort, but as critical and essential inquiry of relevance to us all. The institute is dedicated to redressing these shortcomings by creating a vibrant, worldwide community of scientists, top thinkers, and outreach specialists who tackle deep questions in physics, cosmology, and related fields. FQXi is also committed to engaging with the public and communicating the implications of this foundational research for the growth of human understanding. v
- 10. As part of this endeavor, FQXi organizes an annual essay contest, which is open to everyone, from professional researchers to members of the public. These contests are designed to focus minds and efforts on deep questions that could have a pro- found impact across multiple disciplines. The contest is judged by an expert panel and up to 20 prizes are awarded. Each year, the contest features well over a hundred entries, stimulating ongoing online discussion long after the close of the contest. We are delighted to share this collection, inspired by the 2015 contest, “Trick or Truth: The Mysterious Connection Between Physics and Mathematics.” In line with our desire to bring foundational questions to the widest possible audience, the entries, in their original form, were written in a style that was suitable for the general public. In this book, which is aimed at an interdisciplinary scientific audience, the authors have been invited to expand upon their original essays and include technical details and discussion that may enhance their essays for a more professional readership, while remaining accessible to non-specialists in their field. FQXi would like to thank our contest partners: Nanotronics Imaging, The Peter and Patricia Gruber Foundation, The John Templeton Foundation, and Scientific American. The editors are indebted to FQXi’s scientific director, Max Tegmark, and managing director, Kavita Rajanna, who were instrumental in the development of the contest. We are also grateful to Angela Lahee at Springer for her guidance and support in driving this project forward. 2015 Anthony Aguirre Brendan Foster Zeeya Merali vi Preface
- 11. Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Anthony Aguirre, Brendan Foster and Zeeya Merali Children of the Cosmos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Sylvia Wenmackers Mathematics Is Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 M.S. Leifer My God, It’s Full of Clones: Living in a Mathematical Universe . . . . . . 41 Marc Séguin Let’s Consider Two Spherical Chickens . . . . . . . . . . . . . . . . . . . . . . . . 55 Tommaso Bolognesi The Raven and the Writing Desk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Ian T. Durham The Deeper Roles of Mathematics in Physical Laws . . . . . . . . . . . . . . . 77 Kevin H. Knuth How Mathematics Meets the World . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Tim Maudlin Mathematics: Intuition’s Consistency Check. . . . . . . . . . . . . . . . . . . . . 103 Ken Wharton How Not to Factor a Miracle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Derek K. Wise The Language of Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 David Garfinkle Demystifying the Applicability of Mathematics . . . . . . . . . . . . . . . . . . . 135 Nicolas Fillion vii
- 12. Why Mathematics Works so Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Noson S. Yanofsky Genesis of a Pythagorean Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Alexey Burov and Lev Burov Beyond Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Sophía Magnúsdóttir The Descent of Math. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Sara Imari Walker The Ultimate Tactics of Self-referential Systems . . . . . . . . . . . . . . . . . . 193 Christine C. Dantas Cognitive Science and the Connection Between Physics and Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Anshu Gupta Mujumdar and Tejinder Singh A Universe from Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Philip Gibbs And the Math Will Set You Free . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Ovidiu Cristinel Stoica Appendix: List of Winners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 viii Contents
- 13. Introduction Anthony Aguirre, Brendan Foster and Zeeya Merali The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. Eugene Wigner (1960)1 Modern mathematics is the formal study of structures that can be defined in a purely abstract way, without any human baggage. Think of mathematical symbols as mere labels without intrinsic meaning. It doesn’t matter whether you write “two plus two equals four”, “2 + 2 = 4” or “dos mas dos igual a cuatro”. The notation used to denote the entities and the relations is irrelevant; the only properties of integers are those embodied by the relations between them. That is, we don’t invent mathematical structures – we discover them, and invent only the notation for describing them. Max Tegmark (2014)2 Theoretical physics has developed hand-in-hand with mathematics. It seems almost impossible to imagine describing the fundamental laws of reality without recourse to a mathematical framework; at the same time, questions in physics have inspired many 1Wigner, E. in “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Communications in Pure and Applied Mathematics (John Wiley & Sons: 1960). 2Tegmark, M. in Our Mathematical Universe: My Quest for the Ultimate Nature of Reality (Random House: 2014). A. Aguirre (B) Department of Physics, University of California, Santa Cruz, CA, USA e-mail: aguirre@scipp.ucsc.edu B. Foster · Z. Merali Foundational Questions Institute, Decatur, GA, USA e-mail: foster@fqxi.org Z. Merali e-mail: merali@fqxi.org © Springer International Publishing Switzerland 2016 A. Aguirre et al. (eds.), Trick or Truth?, The Frontiers Collection, DOI 10.1007/978-3-319-27495-9_1 1
- 14. 2 A. Aguirre et al. discoveries in mathematics. In the seventeenth century, for instance, Isaac Newton and Gottfried Wilhelm von Leibniz independently developed calculus, a technique that formed the bedrock of much of Newtonian mechanics and became an essential tool for physicists in the centuries that followed. Newton laid the groundwork for the development of modern theoretical physics as an essentially mathematical discipline. By contrast, mathematics appears to play far less of an integral role in the other sciences. Connections between pure mathematics and the physical world have sometimes only become apparent long after the development of the mathematical techniques, making the link seem even stranger. In the 1930s, for example, physicist Eugene Wigner realised that the abstract mathematics of group and representation theory had direct relevance to particle physics. By uncovering symmetry principles that relate different particles, physicists have since been able to predict the existence of new particles, which were later discovered. In 1960, Wigner wrote an essay pondering such correspondences entitled, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. This issue is also of particular interest to FQXi. One of the Institute’s scientific directors, Max Tegmark, has attempted to explain this intimate relationship by posit- ing that physical reality is not simply represented by mathematics, it is mathematics. This “mathematical universe hypothesis” has been expounded in Tegmark’s book, Our Mathematical Universe: My Quest for the Ultimate Nature of Reality, published in 2014. However, his view remains controversial. So, why does there seem to be a mysterious connection between physics and mathematics? This is the question that FQXi posed in our 2015 essay contest. We asked entrants to consider whether the apparently special relationship between the two disciplines is real or illusory: trick or truth? The contest drew over 200 entries, from thinkers based in 41 countries across 6 continents, both within and outside the academic system. A key aim of these essay contests is to stimulate discussion and the online forum for the contest has generated over 7,000 comments, to date. This volume comprises 19 of the winning essays, which have been expanded and modified from their original forms by the authors, in part, to address points raised by commentators in the forums. InChap.2,ourfirst-prizewinner,SylviaWenmackers,arguesthattheeffectiveness of mathematical models at describing the physical world may not be as unreasonable as it appears at first sight. We must remember, she says, that humans are “children of the cosmos” who have evolved as part of the universe that we seek to describe. When this is taken into account, she says, our ability to model the world is far less strange. Our joint second place winners, Matthew Saul Leifer and Marc Séguin, both wrote essays that explicitly address Tegmark’s assertion that physics is mathemat- ics. In Chap.3, Leifer claims that, on the contrary, mathematics is physics. He explains how this most abstract of disciplines still has its roots in observations of the physical world. In Chap.4, Séguin takes Tegmark’s argument to what he says is its logical conclusion. He states that the hypothesis implies that we must live in a
- 15. Introduction 3 “Maxiverse”—a multitude of universes in which every possible observation happens somewhere—and he outlines its implications. Tommaso Bolognesi also considered Tegmark’s mathematical universe hypothe- sis in his essay. Bolognesi was awarded the prize for “most creative presentation” for a fictional work, reproduced in Chap.5, in which a detective attempts to solve crimes by considering whether there is some truth in the idea that physics is mathemat- ics. Chapter 6 provides another inventive take on the question: Ian Durham presents two characters from Lewis Carroll’s Alice’s Adventures in Wonderland discussing the possibility that there are two realities, representational and tangible, and how to reconcile them. Durham’s essay won FQXi’s “entertainment” prize. TheauthorsofChaps.7–9eachuseconcreteexamplestoexplainwhymathematics has been so successful at describing the physical world, despite being the product of human creativity. Kevin Knuth discusses the derivation of additivity, while Tim Maudlin uses a different mathematical language to explain geometrical structure. Ken Wharton uses the example of the flow of time to demonstrate why branches of mathematics often develop faster than the physical models to which they are applied. A number of entrants addressed Wigner’s original claim concerning the “unrea- sonable effectiveness” of mathematics head on. In Chap.10, Derek Wise argues that Wigner was wrong to assume that mathematics developed entirely independently from the physics that it is used to describe. In Chap.11, David Garfinkle makes a similar case, noting that new mathematics is built on older mathematics, just as new models of physics are modifications of older ones, and that those older branches of physics and mathematics may have been developed in tandem, explaining the seem- ingly surprising connections between more recent developments. Nicolas Fillion, in Chap.12, attempts to demystify the relationship between physics and mathematics by considering how mathematical models are constructed as approximations of real- ity. In Chap.13, Noson Yonofsky argues that the relationship is entirely reasonable if you compare the symmetry principles that underlie both physics and mathematics. While in Chap.14, Sophía Magnúsdóttir takes on the role of a “pragmatic physicist” to make the case that scientific models do not have to be mathematical to be useful. The puzzle of how humans can make sense of physics, at all, is tackled in Chaps.15 and 16. In their essay, Alexey Burov and Lev Burov consider the fine-tuning of the fundamental laws of nature that give rise not only to life, but also make the universe understandable to people. The “out-of-the-box thinking” prize was awarded to Sara Imari Walker who argued that the comprehensibility of the universe is not so astonishing if we accept that the evolution of the structure of reality will favour the development of states that are connected to other existing states in the universe. Some entrants delved more deeply into the relationship between mathematics and the human mind. In Chap.17, Christine Cordula Dantas argues that mathematics is an essential feature of how conscious minds understand themselves. Anshu Gupta Mujumdar and Tejinder Singh were awarded the “creative thinking” prize for their essay, presented in Chap.18, which invokes cognitive science to explain how the abstract features of mathematics have seeds not in some Platonic plane, but within the brain.
- 16. 4 A. Aguirre et al. Finally, in Chaps.19 and 20, Philip Gibbs (the winner of our “non-academic” prize) and Cristinel Stoica use examples from modern physics to review many of the questions surrounding the mysterious connection between mathematics and physics—showing why it is so hard to uncover whether that relationship is a trick or truth. This compilation brings together the writings of professional researchers and non- academics. The contributors to this volume include those trained in mathematics, physics, astronomy, philosophy and computer science. The contest generated some of our most imaginatively structured essays, as entrants strived to solve a mystery about the nature of reality that will no doubt remain with us for centuries to come: Does physics wear mathematics like a costume, or is math a fundamental part of physical reality?
- 17. Children of the Cosmos Presenting a Toy Model of Science with a Supporting Cast of Infinitesimals Sylvia Wenmackers [A]ll our science, measured against reality, is primitive and childlike – and yet it is the most precious thing we have. Albert Einstein [1, p. 404] […] I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. Isaac Newton [2, p. 54] Abstract Mathematics may seem unreasonably effective in the natural sciences, in particular in physics. In this essay, I argue that this judgment can be attributed, at least in part, to selection effects. In support of this central claim, I offer four elements. The first element is that we are creatures that evolved within this Universe, and that our pattern finding abilities are selected by this very environment. The second element is that our mathematics—although not fully constrained by the natural world—is strongly inspired by our perception of it. Related to this, the third element finds fault with the usual assessment of the efficiency of mathematics: our focus on the rare successes leaves us blind to the ubiquitous failures (selection bias). The fourth element is that the act of applying mathematics provides many more degrees of freedom than those internal to mathematics. This final element will be illustrated by the usage of ‘infinitesimals’ in the context of mathematics and that of physics. In 1960, Wigner wrote an article on this topic [4] and many (but not all) later authors have echoed his assessment that the success of mathematics in physics is a mystery. The above quote is attributed to Isaac Newton shortly before his death (so in 1727 our shortly before), from an anecdote in turn attributed to [Andrew Michael] Ramsey by J. Spence [2]. See also footnote 31 in [3]. S. Wenmackers (B) KU Leuven, Centre for Logic and Analytic Philosophy, Institute of Philosophy, Kardinaal Mercierplein 2—Bus 3200, 3000 Leuven, Belgium e-mail: sylvia.wenmackers@hiw.kuleuven.be S. Wenmackers University of Groningen, Faculty of Philosophy, Oude Boteringestraat 52, 9712 GL Groningen, The Netherlands © Springer International Publishing Switzerland 2016 A. Aguirre et al. (eds.), Trick or Truth?, The Frontiers Collection, DOI 10.1007/978-3-319-27495-9_2 5
- 18. 6 S. Wenmackers At the end of this essay, I will revisit Wigner and three earlier replies that harmonize with my own view. I will also explore some of Einstein’s ideas that are connected to this. But first, I briefly expose my views of science and mathematics, since these form the canvass of my central claim. Toy Model of Science Science can be viewed as a long-lasting and collective attempt at assembling an enormous jigsaw puzzle. The pieces of the puzzle consist of our experiences (in par- ticular those that are intersubjectively verifiable) and our argumentations about them (often in the form of mathematical models and theories). The search for additional pieces is part of the game. Any piece that we add to the puzzle at one time may be removed later on. Nobody knows how many pieces there are, what the shape of the border looks like, or whether the pieces belong to the same puzzle at all. We assume optimistically that this is the case, indeed, and we attempt to connect all the pieces of the puzzle that have been placed on the table so far.1 So, like Einstein and (allegedly) Newton in the quotes appearing on the title page, I view science as a playful and limited activity, which is at the same time a highly valuable and unprecedented one. Scientific knowledge is fallible, but there is no better way to obtain knowledge. Hence, it seems wise to base our other epistemic endeavors (such as philosophy) on science—a position known as ‘naturalism’. In addition, there is no more secure foundation for scientific knowledge beyond science itself. The epistemic position of ‘coherentism’ lends support to the positive and optimistic project of science. It has been phrased most evocatively by Otto Neurath [7, p. 206]: Like sailors we are, who must rebuild their ship upon the open sea, without ever being able to put it in a dockyard to dismantle it and to reconstruct it from the best materials.2 We are in the middle of something and we are not granted the luxury of a fresh start. Hence, we cannot analyze the apparent unreasonable effectiveness of mathematics in science from any better starting point either. Condemned we are to deciphering the issue from the incomplete picture that emerges from the scientific puzzle itself, while its pieces keep moving. A dizzying experience. Since I mentioned “mathematical models and theories”, I should also express my view on those.3 To me, mathematics is a long-lasting and collective attempt at 1Making these connections involves developing narratives. Ultimately, science is about storytelling. “The anthropologists got it wrong when they named our species Homo sapiens (‘wise man’). In any case it’s an arrogant and bigheaded thing to say, wisdom being one of our least evident features. In reality, we are Pan narrans, the storytelling chimpanzee.”—Ian Stewart, Jack Cohen, and Terry Pratchett (2002) [6, p. 32]. 2This is my translation of the German quote [7, p. 206]: “Wie Schiffer sind wir, die ihr Schiff auf offener See umbauen müssen, ohne es jemals in einem Dock zerlegen und aus besten Bestandteilen neu errichten zu können.” 3In the current context, I differentiate little between ‘models’ and ‘theories’. For a more detailed account of scientific models, see [5].
- 19. Children of the Cosmos 7 thinking systematically about hypothetical structures—or imaginary puzzles, if you like. (More on this in section “Mathematics as Constrained Imagination” below.) Selection Effects Behind Perceived Effectiveness of Mathematics in Physics The four elements brought to the fore in this section collectively support my defla- tionary conclusion, that the effectiveness of mathematics is neither very surprising nor unreasonable. A Natural History of Mathematicians This section addresses the two following questions. What enables us to do mathemat- ics at all? And how is it that we cannot simply describe real-world phenomena with mathematics, but even predict later observations with it? I think that we throw dust in our own eyes if we do not take into account to which high degree we—as a biological species, including our cognitive abilities that allow us to develop mathematics—have been selected by this reality. To address the matter of whether mathematical success in physics is trick or truth (or something else), and in the spirit of naturalism and coherentism (section “Toy Model of Science”), we need to connect different pieces of the scientific puzzle. In the ancient Greek era, the number of available pieces was substantially smaller than it is now. Plato was amongst the first to postulate parallel worlds: alongside our concrete world, populated by imperfect particulars, he postulated a world of universal Ideas or ideal Forms, amongst which the mathematical Ideas sat on their thrones of abstract existence.4 In this view, our material world is merely an imperfect shadow of the word of perfect Forms. Our knowledge of mathematics is then attributed to our soul’s memories from a happier time, at which it had not yet been incarcerated in a body and its vista had not yet been limited by our unreliable senses. This grand vision of an abstract world beyond our own has crippled natural phi- losophy ever since. The time has come to lay this view to rest and to search for better answers, guided by science. Although large parts of the scientific puzzle remain miss- ing in our time, I do think that we are in a better position than the ancient Athenian scholars to descry the contours of an answer to the questions posed at the beginning of this section. Let us first take stock of what is on the table concerning the origin of our math- ematical knowledge. Is mathematical knowledge innate, as Plato’s view implied? According to current science, the matter is a bit more subtle: mathematical knowledge 4I will have more to say on the ancient Greek view on mathematics and science in section “A Speculative Question Concerning the Unthinkable”.
- 20. 8 S. Wenmackers is not innate (unfortunately, since otherwise we would not have such a hard time learning or teaching mathematics), but there are robust findings that very young chil- dren (as well as newborns of non-human animals, for that matter) possess numerical abilities [8, 9]. So, we have innate cognitive abilities, that allow us to learn how to count and—with further effort—to study and to develop more abstract forms of mathematics. This raises the further question as to the origin of these abilities. To answer it, we rely on the coherent picture of science, which tells us this: if our senses and reasoning did not work at all, at least to an approximation sufficient for survival, our ancestors would not have survived long enough to raise offspring and we would not have come into being. Among the traits that have been selected, our ancestors passed on to us certain cognitive abilities (as well as associated vices: more on this below). On this view, we owe our innate numerical abilities to the biological evolution of our species and its predecessors. Let me give a number of examples to illustrate how our proto-mathematical capac- ities might have been useful in earlier evolutionary stages of our species. Being able to estimate and to compare the number of fruits hanging from different trees contributes to efficient foraging patterns. So does the recognition of regional and seasonal5 pat- terns in the fruition of plants and the migration of animals. And the ability to plan future actions (rather than only being able to react to immediate incentives) requires a crude form of extrapolation of past observations. These traits, which turned out to be advantageous during evolution, lie at the basis of our current power to think abstractly and to act with foresight. Our current abilities are advanced, yet limited. Let us first assess our extrapolative capacities: we are far from perfect predictors of the future. Sometimes, we fail to take into account factors that are relevant, or we are faced with deterministic, yet intrinsically chaotic systems. Consider, for example, a solar eclipse. An impending occultation is predicted many years ahead. However, whether the weather will be such that we can view the phenomenon from a particular position on the Earth’s surface, that is something we cannot predict reliably a week ahead. Let us then turn to the more basic cognitive faculty of recognizing patterns. We are prone to patternicity, which is a bias that makes us see patterns in accidental correlations [10]. This patternicity also explains why we like to play ‘connect the dots’ while looking at the night sky: our brains are wired to see patterns in the stars, even though the objects we thus group into constellations are typically not in each other’s vicinity; the patterns are merely apparent from our earthbound position. In our evolutionary past, appropriately identifying many patterns yielded a larger advantage than the disadvantage due to false positives. In the case of a tiger, it is clear that one false negative can be lethal. But increasing appropriate positives invariable comes at the cost of increasing false positives as well.6 As a species, we must make do without venom or an exoskeleton, alas, but we have higher cognitive abilities that allow us to plan our actions and to devise mathematics. 5Or ‘spatiotemporal’, if you like to talk like a physicist. 6The same trade-off occurs, for instance, in medical testing and law cases.
- 21. Children of the Cosmos 9 These are our key traits for survival (although past success does not guarantee our future-proofness). In sum, mathematics is a form of human reasoning—the most sophisticated of its kind. When this reasoning is combined with empirical facts, we should not be perplexed that—on occasions—this allows us to effectively describe and even predict features of the natural world. The fact that our reasoning can be applied successfully to this aim is precisely why the traits that enable us to achieve this were selected in our biological evolution. Mathematics as Constrained Imagination In my view, mathematics is about exploring hypothetical structures; some call it the science of patterns. Where do these structures or patterns come from? Well, they may be direct abstractions of objects or processes in reality, but they may also be inspired by reality in a more indirect fashion. For instance, we could start from an abstraction of an actual object or process, only to negate one or more of its properties—just think of mathematics’ ongoing obsession with the infinite (literally the non-finite). Examples involving such an explicit negation clearly demonstrate that the goal of mathematics is not representation of the real world or advancing natural science. Nevertheless, this playful and free exercise in pure mathematics may—initially unin- tended and finally unexpected—turn out to be applicable to abstractions of objects and processes in reality (completely different from the one we started from). Stated in this way, the effectiveness of mathematics surely seems unreasonable. However, I argue that there are additional factors at play that can explain this success—making these unintentional applications of mathematics more likely after all. Let us return to the toy metaphor, assuming, for definiteness, that the puzzle of nat- ural science appears to be a planar one. Of course, this is no reason for mathematicians not to think up higher dimensional puzzles, since their activity is merely imaginative play, unhindered by any of the empirical jigsaw pieces. However, it is plausible that the initial inspiration for considering, say, toroidal or hypercubic puzzles has been prompted by difficulties with fitting the empirical pieces into a planar configuration.7 In addition, and irrespective of its source, this merely mathematical construct may subsequently prompt speculations about the status of the scientific puzzle. Due to feedback processes like these, the imaginative play is not as unconstrained as we might have assumed at the outset. The hypothetical structures of mathematics are not concocted in a physical or conceptual vacuum. Even in pure mathematics, this physical selection bias acts very closely to the source of innovation and creativity. In the previous section, I highlighted that humans, including mathematicians, have evolved in this Universe. Mathematics itself also evolves by considering variations on earlier ideas and selection: this is a form of cultural evolution which allows changes on a much shorter time scale than biological evolution does. Just like in biology, this 7In this example, considering the negation of the planar assumption—rather than any of the other background assumptions—is prompted by troubles in physics.
- 22. 10 S. Wenmackers variation produces many unviable results. Evolution is squandermanious—quite the opposite of efficient. The selection process is mainly driven by cultural factors, which are internal to mathematics (favoring theories that exhibit epistemic virtues such as beauty and simplicity). But, as we saw in the previous paragraph, empirical factors come into play as well, mediated by external interactions with science. Although mathematics is often described as an a priori activity, unstained by any empirical input, this description itself involves an idealization. In reality, there is no a priori. Mathematics Fails Science More Often Than Not For each abstraction, many variations are possible, the majority of which are not applicable to our world in any way. The effectiveness perceived by Wigner [4] may be due to yet another form of selection bias: one that makes us prone to focus on the winners, not the bad shots. Moreover, even scientific applications of mathematics that are widely considered to be highly successful have a limited range of applicability and even within that range they have a limited accuracy. Among the mathematics books in university libraries, many are filled with theories for which not a single real world application has been found.8 We could measure the efficiency of mathematics for the natural science as follows: divide the number of pages that contain scientifically applicable results by the total number of pages produced in pure mathematics. My conjecture is that, on this definition, the efficiency is very low. In the previous section we saw that research, even in pure mathematics, is biased towards the themes of the natural sciences. If we take this into account, the effectiveness of mathematics in the natural sciences does come out as unreasonable— unreasonably low, that is.9 Maybe it was unfair to focus on pure mathematics in the proposed definition for efficiency? A large part of the current mathematical corpus deals with applied mathematics,fromdifferentialequationstobio-statistics.Ifwemeasuretheefficiency by dividing the number of ‘applicable pages’ by the total number of pages produced in all branches of mathematics, we certainly get a much higher percentage. But, now, the effectiveness of mathematics in the natural sciences appears reasonable enough, sinceresearchandpublicationsinappliedmathematicsare(rightfully)biasedtowards real world applicability. At this point, you may object that Wigner made a categorical point that there is some part of mathematics at all that works well, even if this does not constitute all or most of mathematics. I am sympathetic to this objection (and the current point is the least important one in my argument), but then what is the contrasting case: that 8This is fine, of course, since this is not the goal of mathematics. 9Here, I recommend humming a Shania Twain song: “So, you’re a rocket scientist. That don’t impress me much.” If you are too young to know this song, consult your inner teenager for the appropriate dose of underwhelmedness.
- 23. Children of the Cosmos 11 no mathematics would describe anything in the Universe? I offer some speculations about this in section “A Speculative Question Concerning the Unthinkable”. Abundant Degrees of Freedom in Applying Mathematics: The Case of Infinitesimals I once attended a lecture in which the speaker claimed that “There is a matter of fact about how many people are in this room”. Unbeknownst to anyone else in that room, I was pregnant at the time, and I was unsure whether an unborn child should be included in the number of people or not. To me, examples like this show that we can apply mathematically crisp concepts (such as the counting numbers) to the world, but only because other concepts (like person or atom) are sufficiently vague. The natural sciences aim to formulate their theories in a mathematically precise way, so it seems fitting to call them the ‘exact sciences’. However, the natural sci- ences also allow—and often require—deviations from full mathematical rigor. Many practices that are acceptable to physicists—such as order of magnitude calculations, estimations of errors, and loose talk involving infinitesimals—are frowned upon by mathematicians. Moreover, all our empirical methods have a limited range and sen- sitivity, so all experiments give rise to measurement errors. Viewed as such, one may deny that any empirical science can be fully exact. In particular, systematic discrep- ancies between our models and the actual world can remain hidden for a long time, provided that the effects are sufficiently small, compared to our current background theories and empirical techniques. Einstein put it like this: “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality” [11, p. 28]. To illustrate this point, I will concentrate on the calculus—the mathematics of differ- ential and integral equations—and consider the role of infinitesimals in mathematics as well as in physics. In mathematics, infinitesimals played an important role during the development of the calculus, especially in the work of Leibniz [12], but also in that of Newton (where they figure as ‘evanescent increments’) [13]. The development of the infinitesimal calculus was motivated by physics: geometric problems in the context of optics, as well as dynamical problems involving rates of change. Berkeley [14] ridiculed these analysts as employing “ghosts of departed quantities”. It has taken a long time to find a consistent definition of infinitesimals that holds up to the current standards of mathematical rigour, but meanwhile this has been achieved [15]. The contemporary definition of infinitesimals considers them in the context of an incomplete, ordered field of ‘hyperreal’ numbers, which is non-Archimedean: unlike the field of real numbers, it does contain non-zero, yet infinitely small numbers (infinitesimals).10 The alternative calculus based on hyperreal numbers, called ‘non-standard analysis’ 10Here, I mean by infinitesimals numbers larger than zero, yet smaller than 1/n for any natural number n.
- 24. 12 S. Wenmackers (NSA), is conceptually closer to Leibniz’s original work (as compared to standard analysis). While infinitesimals have long been banned from mathematics, they remained in fashion within the sciences, in particular in physics: not only in informal discourse, but also in didactics, explanations, and qualitative reasoning. It has been suggested that NSA can provide a post hoc justification for how infinitesimals are used in physics [16]. Indeed, NSA seems a very appealing framework for theoretical physics: it respects how physicists are already thinking of derivatives, differential equations, series expansions, and the like, and it is fully rigorous.11 Rephrasing old results in the language of NSA may yield new insights. For instance, NSA can be employed to make sense of classical limits in physics: classi- cal mechanics can be modelled as quantum mechanics with an infinitesimal Planck constant [22]. Likewise, Newtonian mechanics can be modelled as a relativity theory with an infinite maximal speed, c (or infinitesimal 1/c). Infinitesimal numbers are indistinguishable from zero (within the real numbers), yet distinct from zero (as can be made explicit in the hyperreal numbers). This is suggestive of a physical interpretation of infinitesimals as ‘currently unobserv- able quantities’. The ontological status of unobservables is an important issue in the realism–anti-realism debate [23]. Whereas constructive empiricists interpret ‘observ- ability’ as ‘detectability by the human, unaided senses’ [24], realists regard ‘observ- ability’ as a vague, context-dependent notion [25]. When an apparatus with better resolving power is developed, some quantities that used to be unobservably small become observable [26, 27]. This shift in the observable-unobservable distinction can be modelled by a form of NSA, called relative analysis, as a move to a finer context level [28]. Doing so requires the existing static theory to be extended by new principles that constrain the allowable dynamics [29]. The interpretation of (relative) infinitesimals as (currently) unobservable quanti- ties is suggestive of why the calculus is so applicable to the natural sciences: it appears that infinitesimals provide scientists with the flexibility they need to fit mathematical theories to the empirically accessible world. To return to the jigsaw puzzle analogy of section “Toy Model of Science”: we need some tolerance at the edges of the pieces. If the fit is too tight, it becomes impossible to connect them at all. A Speculative Question Concerning the Unthinkable Could our cosmos have been different—so different that a mathematical description of it would have been fundamentally impossible (irrespective of whether life could 11It hasbeenshownthat physical problemscanbe rephrasedintermsofNSA[17],bothinthe context of classical physics (Lagrangian mechanics [18]) and of quantum mechanics (quantum field theory [19], spin models [20], relativistic quantum mechanics [21], and scattering [18]). Apart from formal aspects (mathematical rigour), such a translation also offers more substantial advantages, such as easier (shorter) proofs.
- 25. Children of the Cosmos 13 emerge in it)? Some readers may have the impression that I have merely explored issues in the vicinity of this mystery, without addressing it directly. Before I indulge in this speculation, it may be worthwhile to remember that the very notion of a ‘cosmos’ emerged in ancient Greek philosophy, with the school of Pythagoras, where it referred to the order of the Universe (not the Universe itself). It is closely related to the search for archai or fundamental ordering principles. It is well known that the Pythagorians took the whole numbers and—by extension— mathematics as the ordering principle of the Universe. Their speculations about a mathematically harmonious music of the spheres resonated with Plato and Johannes Kepler (the great astronomer, but also the last great neoplatonist). Since these archai had to be understandable to humans, without divine intervention or mystical revela- tion, they had to be limited in number and sufficiently simple. So, the idea that the laws of nature have to be such that they can be printed on the front of a T-shirt, goes back to long before the invention of the T-shirt.12 In this sense, the answer to the speculative question at the start of this section is ‘no’ and trivially so, for otherwise it would not be a cosmos. Yet, even if we understand ‘our cosmos’ as ‘the Universe’, there is a strong cultural bias to answer the speculative question in the negative. In section “A Natural History of Mathematicians”, I considered our proto- mathematical abilities as well as their limits. At least in some areas, our predic- tions do better than mere guesses. This strongly suggests that there are patterns in the world itself—maybe not the patterns that we ascribe to it, since these may fail, but patterns all the same. It is then often taken to be self-evident that these patterns must be mathematical, but to me this is a substantial additional assumption. On my view of mathematics, the further step amounts to claiming that nature itself is—at least in principle—understandable by humans. I think that all we understand about nature are our mathematical representations of it.13 Ultimately, reality is not something to be understood, merely to be. (And, for us, to be part of.) When we try to imagine a world that would defy our mathematical prowess, it is temptingtothinkofaworldthatistotallyrandom.However,thisattemptisfutile.Pure randomness is a human idealization of maximally unpredictable outcomes (like a perfectly fair lottery [30]). Yet, random processes are very well-behaved: they consist of events that may be maximally unpredictable in isolation, but collectively they produce strong regularities. It is no longer a mystery to us how order emerges from chaos. In fact, we have entire fields of mathematics for that, called probability theory and statistics, which are closely related to branches of physics, such as statistical mechanics. As a second attempt, we could propose a Daliesque world, in which elements combine in unprecedented ways and the logic seems to change midgame: rigid clocks become fluid, elephants get stilts, and tigers emerge from the mouths of fish shooting 12In case this remark made you wonder: the T-shirt was invented about a century ago. 13My view of mathematics might raise the question: “Why, then, should we expect that anything as human and abstract as mathematics applies to concrete reality?” I think this question is based on a false assumption, due to prolonged exposure to Platonism—remnants of which are abundant in our culture.
- 26. 14 S. Wenmackers from a pomegranate. Yet, even such surrealistic tableaus have meta-regularities of their own. Many people are able to recognize a Dalí painting instantly as his work, even if they have not seen this particular painting before. Since we started from human works of art, unsurprisingly, the strategy fails to outpace our own constrained imagination. At best, I can imagine a world in which processes cannot be summarized or approximated in a meaningful way. Our form of intelligence is aimed at finding the gist in information streams, so it would not help us in this world (in which it would not arise spontaneously by biological evolution either). In any case, what I can imagine about such a world remains very vague—insufficient for any mathematical description. Maybe there are better proposals out there? Max Tegmark has put forward an evocative picture of the ultimate multiverse as consisting of all the orderings that are mathematically possible [31]. (See also Marc Séguin’s contribution [32].) Surely, this constitutes a luscious multiplicity. From my view of mathematics as constrained imagination, however, the idea of a mathematical multiverse is still restricted by what is thinkable by us, humans. Aristotle described us as thinking animals, but for the current purpose ‘mathematizing mammals’ may fit even better. My diagnosis of the situation is that the speculative question asks us to boldly go even beyond Tegmark’s multiverse and thus to exceed the limits of our cognitive kung fu: even with mathematics, we cannot think the unthinkable. Reflections on Wigner and Einstein In this section, I return to Wigner [4] and compare my own reflections to earlier replies by Hamming [33], Grattan-Guinness [34], and Abbott [35]. I end with a short reflection on some ideas of Einstein [11] that predate Wigner’s article by four decades. Wigner’s Two Miracles Wigner wrote about “two miracles”: “the existence of the laws of nature” and “the human mind’s capacity to divine them” [4, p. 7]. First and foremost, I hope that my essay helps to see that we need not presuppose the former to understand the latter: it is by assuming that the Universe forms a cosmos that we have started reading laws into it. Galileo Galilei later told us that those laws are mathematical ones. The very term “laws of nature” may be misleading and for this reason, I avoided it so far (except for the remark of fitting them on a T-shirt). The fact that our so-called laws can be expressed with the help of mathematics should be telling, since that is our science of patterns. When we open Galileo’s proverbial book of nature, we find it filled with our own handwriting.
- 27. Children of the Cosmos 15 To illustrate the unreasonable effectiveness of mathematics, Wigner offered the following analogy for it: consider a man with many keys in front of many doors, who “always hits on the right key on the first or second try” [4, p. 2]. Lucky streaks like this may seem to require further explanation. However, if there are many people, each with many keys, it becomes likely that at least one of them will have an experience like Wigner’s man—and no further explanation is needed (see also Hand [36]). There are indeed many people active in mathematics and science, and few of them succeed “on the first or second try”—or at all. In the essay, I argued that the perceived effectiveness of mathematics in physics can be diagnosed in terms of selection bias. The same applies to the metaphor Wigner presented. Wigner found it hard to believe that the perfection of our reasoning power was brought about by Darwin’s process of natural selection [4, p. 3]. Ironically, the selection bias that he may have fallen prey to is a good illustration of the lack of perfection of our reasoning powers. To be clear, I do not claim that Darwin’s theory ofbiologicalevolutionsufficestoexplainthesuccessof(certainpartsof)mathematics in physics. However, a similar combination of variation and selection is at work in the evolution of mathematics and science. See also Pólya, as cited by Grattan-Guinness, who has given an iterative or evolutionary description of the development of science [37, Vol. 2, p. 158]. Previous Replies to Wigner In his description of mathematics, Wigner wrote about “defining concepts beyond those contained in the axioms” [4, p. 3]. Wigner did not, however, reflect on where those axioms come from in the first place. This has been criticized by Hamming [33], a mathematician who worked on the Manhattan Project and for Bell Labs. Axioms or postulates are not specified upfront; instead, mathematicians may try various pos- tulates until theorems follow that harmonize with their initial vague ideas. Hamming cited the Pythagorean theorem and Cauchy’s theorem as examples: if mathemati- cians would have started out with a system in which those crucial results would not hold, then—according to Hamming—they would have changed their postulates until they did. And, of course, the initial vague ideas are thoughts produced by beings entrenched in the physical world. This brings us to Putnam, who pointed out that mathematical knowledge resembles empirical knowledge in many respects: “the cri- terion of success in mathematics is the success of its ideas in practice” [38, p. 529]. Wigner did concede that not any mathematical concept will do for the formulation of laws of nature in physics [4, p. 7], but he claimed that “in many if not most cases” these concepts were developed “independently by the physicist and recognized then as having been conceived before by the mathematician” [4, p. 7] (my emphasis). I think this part is misleading: there is a lot of interaction between mathematics and physics and what is being ‘recognized’ is actually the finding of a new analogy. We may think that we are merely discovering a similarity, when we are really forging new connections, which may subtly alter both sides. This is a creative element of
- 28. 16 S. Wenmackers great importance within mathematics as well as in finding applications to other fields, in which our patternicity may be a virtue rather than a vice. Both aspects have been illuminated by Grattan-Guinness, a historian of mathe- matics, who argued that “much mathematics has been motivated by interpretations in the sciences” [34, p. 7]. He stressed the importance of analogies within mathematics and between mathematics and natural science and he gave historical examples in which mathematics and physics take turns in reshaping earlier concepts. Moreover, he remarked that there are many analogies that can be tried (somewhat similar to the ideas in section “Abundant Degrees of Freedom in Applying Mathematics: The Case of Infinitesimals”), but that only the successful ones are taken into account when assessing the effectiveness of mathematics (postselection as in section “Mathematics as Constrained Imagination”). My essay mainly focused on elementary mathematics and simple models. Of course, there are very complicated mathematical theories in use in advanced physics. In relation to this, Grattan-Guinness observed that “by around 1900 linearisation had become something of a fixation” [34, p. 11], but he also discussed the subsequent “desimplification” or “putting back in the theory effects and factors that had been deliberately left out” [34, p. 13]. In light of my discussion of infinitesimals in this essay, it is curious to observe that Grattan-Guinness and Hamming referred to them too. Grattan-Guinness spoke approvingly of the Leibniz-Euler approach to the calculus because it “often has a better analogy content to the scientific context” [34, p. 15]. Hamming even mentioned NSA, but only as an example of the observation that “logicians can make postulates that put still further entities on the real line” [33, p. 85]. Recently, the reply by Hamming has been developed further by Abbott, a professor in electrical engineering [35]. Whereas Hamming described his recurrent feeling that “God made the universe out of complex numbers” [33, p. 85], Abbott described the complex numbers as “simply a convenience for describing rotations”, concluding that “the ubiquity of complex numbers is not magical at all” [35, p. 2148]. More specifically, Abbott adds two points to Hamming’s earlier observations. Abbott’s first addition is that “all physical laws and mathematical expressions of those laws are […] necessarily compressed due to the limitations of the human mind” [35, p. 2150]. He explains that the associated loss of information does not preclude usefulness “provided the effects we have neglected are small”, which lends itself perfectly to a rephrasing in terms of ‘my’ relative infinitesimals. Abbott’s second addition is that “the class of successful mathematical models is preselected”, which he described as a “Darwinian selection process” [35, p. 2150]. Like I did here, Abbott warned his readers not to overstate the effectiveness of mathematics. Moreover, as an engineer, he is well aware that “when analytical methods become too complex, we simply resort to empirical models and simulations” [35, p. 2148]. The title of Abbot’s piece, “The reasonable ineffectiveness of mathematics”, and the general anti-Platonist stance agree with the views exposed in the current essay. In addition, Abbott tried to show that this debate is relevant, even for those who prefer to “shut up and calculate”, because “there is greater freedom of thought, once we realize that mathematics is something we entirely invent as we go along” [35, p. 2152].
- 29. Children of the Cosmos 17 Einstein’s Philosophy of Science In 1921, so almost forty years before Wigner wrote his article, Einstein gave an address to the Prussian Academy of Sciences in Berlin titled “Geometrie und Erfahrung”. The expanded and translated version contains the following passage (which includes the sentence already quoted in section “Abundant Degrees of Freedom in Applying Mathematics: The Case of Infinitesimals”): At this point an enigma presents itself which in all ages has agitated inquiring minds. How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things. In my opinion the answer to this question is, briefly, this: As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. [11, p. 28] On the one hand, we read of “an enigma”, which seems to set the stage for Wigner’s later question. On the other hand, we are not dealing with a Platonist conception of mathematics here, since Einstein describes it as “a product of human thought”. Yet, how are we to understand Einstein’s addition that mathematics is “independent of experience”? This becomes clearer in the remainder of his text: geometry stems from empirical “earth-measuring”, but modern axiomatic geometry, which allows us to consider multiple axiomatisations (including non-Euclidean ones), remains silent on whether any of these axiom schemes applies to reality [11, pp. 28–32]. Einstein refers approvingly to Schlick’s view that axioms in the modern sense function as implicit definitions, which is in agreement with Hilbert’s formalist position.14 Regarding his own answer to the question, Einstein further explains that axiomatic geometry can be supplemented with a proposition to relate mathematical concepts to objects of experience: “Geometry thus completed is evidently a natural science” [11, p. 32]. My essay agrees with Einstein’s answer: the application of mathematics supplies additional degrees of freedom external to mathematics and we can never be sure that the match is perfect, since empirical precision is always limited. On other occasions, Einstein also pointed out that Kant’s a priori would better be understood as ‘conventional’: a position close to that of Pierre Duhem (see for instance [39]). In my late teens, I read a Dutch translation of essays by Einstein. The oldest essay in that collection stemmed from 1936 [40], so the text from which I quoted does not appear in that book. However, it is clear that similar ideas have influenced me in my formative years. It was a pleasure to reexamine some of them here. They contributed to my decision to become a physicist and a philosopher of physics, which in turn helped me to write this essay. Hence, the appearance of a text in this collection that is in agreement with some of Einstein’s view on science may be a selection effect as well. 14For the influence of Moritz Schlick on Einstein’s ideas, see [39].
- 30. Discovering Diverse Content Through Random Scribd Documents
- 31. 302 The biographies of Francis Xavier are naturally of a kind to excite the critical instincts of a scholar. They are, from the original life by Torsellini, to the latest Jesuit compilation, remarkable for their enthusiasm and unlimited credulity. It is only in such calmer treatises as those of Nicolini, Stephen, Venn, and others, that we get the more just conception of his character. But to be entirely fair to him we should take him from the picture painted by his co-religionists, refusing only those things which are manifestly incongruous or absurd. The work of Bartoli and Maffei may, for example, be regarded as entirely safe in its general statements. From the portraits left to us and preserved in the pages of Nicolini and Mrs. Jameson, we derive a vivid impression of the man’s personal intensity. His eyes are deep and thoughtful; his nose strong, rather blunt, and withal sagacious; and his face is that of a mystic. He is usually represented as gazing upward in religious rapture and his lips are parted. His features are more rugged and forcible than refined. They indicate a rude strength of body and of will rather than a delicate and sensitive nature. Should we have met him personally, he would have given us the impression of an enthusiast, deeply affectionate and profoundly loyal to anything like a military organization. These opinions would have been approved by the fact. We read that his parents desired to educate him as a cavalier, and that he was at first instructed at home in the usual topics. But as he showed zeal and intelligence he was sent, in his eighteenth year, to the College of Ste. Barbe at Paris. Here he completed the study of philosophy, received the degree of Master, and began to give instruction to others. His most intimate friend was Peter Faber, afterward to become one of the earliest adherents of Ignatius Loyola. And the biographers are unwearied in their eulogy of Xavier’s and Faber’s purity of life and morals in the midst of the great temptations of a corrupt city.
- 32. 303 To these two young men, ardent of mind and eager in their ambition, now enters the influence which shapes their destiny. Faber was a Savoyard, poor and of humble birth, while Xavier was well-to- do and possessed the haughty spirit of a Spanish grandee. They were, however, kindling each other up to some scheme of future glory when Ignatius Loyola made his way to Paris. He had been converted a few years before this and had already begun to gather proselytes to his opinions. His purpose in visiting Paris was not merely to avail himself of better facilities for study, but also to secure more followers. It is not strange to us that Loyola, with his great sagacity, should have singled out the two companions and have set himself to win them. Faber’s allegiance, indeed, it was an easy matter to obtain. But Xavier did not so readily fall in with the wishes of the great general of the Jesuits. Faber’s conversion was rapidly accomplished. He was supplied with the Spiritual Exercises, which is, of all books, the best adapted to produce the proper self-abandonment and plastic condition of soul which befit the neophyte of the Society of Jesus. And this work, composed, say the Roman Catholic authorities, in the cavern of Manresa with the help of the Virgin Mary, may be regarded as the keenest instrument by which men’s lives were ever carved into the patterns designed by a superior will. We have no space for a discussion of Jesuitism further than to indicate its methods when they affect the subject before us, but Faber’s behavior undoubtedly had its weight upon Xavier. The Savoyard took to fasting with a perfect fury. In his debilitated condition he was the fit vehicle for spiritual impressions, for ecstasies, and for mystical dreams. He would kneel in the open court in the snow, and sometimes allow himself to be covered with icicles. His bundle of fuel he made into a bed and slept upon it for the few hours of what one biography “scarcely knows whether to call torture or repose.” In fact, he so outran the instruction of Loyola, that that keen observer checked him and prevented what would have reacted against his own designs. “For,” saith quaint Matthew Henry, speaking of another
- 33. 304 subject, “there is a great deal of doing which, by overdoing, is altogether undone.” Xavier was, however, more important to Loyola than Faber. And Xavier was of tougher material and harder to reach. Upon him the intense Loyola bent the blow-pipe flame of his own spirit. He had failed to touch him by texts or by austerities. He therefore changed his tactics altogether and began to soften him by praise, by judicious cultivation of his sympathies, by procuring new scholars for him, and even by attending his lectures and feigning a deep interest in whatever he did. In short, he applied flattery and deference in such a way that he insinuated himself very soon into the confidence of Xavier, and allowed the haughty Don to recognize the high birth and good breeding which he could also claim. This was a master stroke. Faber was after all only a Savoyard; but Loyola was born in a castle, had been a page at the court of Ferdinand, and had led soldiers into the deadliest places of battle. He had also the advantage of being Xavier’s senior by fully fourteen years, for his birth had been contemporaneous with Columbus’s expedition in search of the new world. Here, then, the influence of this strong, undaunted, unflinching spirit began to focus itself upon the young teacher of philosophy. “Resistance to praise,” says the bitter La Rochefoucauld, “is a desire to be praised twice.” And to so acute a student of human nature as Loyola it soon grew evident that he was making progress. This was proved even by the modesty of Xavier. Therefore he redoubled his energies and utilized that marvellous power of adaptation, which was his chief legacy to his order, in obtaining a definite result. He gained ground so fast that Michael Navarro, a faithful servant of the young scholar, became determined to break off this dangerous fascination, and even attempted to kill Loyola in his private apartments. But he, too, was dealing with a brain which never relaxed its vigilance and with a magnetic personality which felt a danger, and moved safely, cat-like, through the dark. He was halted and challenged by the man he came to kill, and being
- 34. crushed down in confusion was thereupon treated with magnanimity, and went away revolving many things in his mind. This was the power of Loyola—a power which sprang, first of all, from his peculiar constitution, and, second, from his fanatical ambition. It has been the key by which the Jesuit has ever since unlocked the doors of palaces and contrived to whisper in the ears of kings. Its extent has been that of the civilized and uncivilized world. In the matter of organization no human fraternity has ever equalled the Society of Jesus. The germs which we behold at Ste. Barbe in Paris have grown into a tree whose roots have taken hold on every soil, and whose fruit has dropped in every clime. The order has invariably employed strategy, intrigue, ingenuity, and perfect combination to secure its ends. It is, as a system, far from being either dead or insignificant. And its real vitality has always sprung from its maxim that its associated members, vowed to celibacy and to the accomplishment of its purposes, should be Perinde ac si cadavera—absolutely subordinate and dead to any other will—in the hands of the “general” who is at the head of its affairs. It has worked, first for itself, second for the Roman Catholic Church, and third for the proselytizing of the heathen and the heretics. It has never neglected to procure in every manner the information it needed to the full extent or to employ its principle that the end to be gained justifies the means that are taken to gain it. Thus it is the legitimate outgrowth of the soldier-courtier-fanatic mind of its founder. And this was the mind which was now spending its splendid resources upon Xavier, playing with him like a trout upon the hook, until it should land him, a completely surrendered man, within its own control. In another sphere and under other influences, Xavier might have been a far different person. He, at least, was sincere in his devotion to the cause. He identified Jesuitism with Christianity and Loyola with Jesus Himself. Hence his character and labors have blinded many persons to the methods which he used and to the results which he sought.
- 35. 305 It must be sufficient for us that Ignatius Loyola had now gotten the mastery of Francis Xavier so perfectly that he could be “applied to the Spiritual Exercises, the furnace in which he [Loyola] was accustomed to refine and purify his chosen vessels.” A sister of the future missionary had become one of the Barefooted Clares, and had aided in dissuading her father from interference. And now we behold Xavier praying with hands and feet tightly bound by cords; or journeying with similar cords about his arms and the calves of his legs until inflammation and ulceration ensued. There were now nine of these converts, but this man outdid the others in his austerities, and finally travelled on foot with them to meet Loyola at Venice in 1537. The society had really been formed on August 15th, 1534, at Montmartre near Paris, and this was but its natural outward movement. At Venice, on January 8th, 1537, they again met their leader and were assigned for duty to the two hospitals of the city. That of the “Incurables” fell to Xavier’s share, and we read that with the morbid devotion characteristic of a devout student of the Exercises, he determined now to conquer his natural repugnance to disease. In the course of his duties he had an unusually hideous ulcer to dress for one of the patients. And the authentic history relates that “encouraging himself to the utmost, he stooped down, kissed the pestilent cancer, licked it several times with his tongue, and finally sucked out the virulent matter to the last drop.” (Bartoli and Maffei, p. 35.) There could be nothing worse than that certainly. And a man who had resolutely sounded this deepest abyss of self-abandonment was marked for the highest honor that the new society could bestow. We cannot doubt Xavier’s sincerity, but the gigantic horror of this performance is of a sort to place the man who has achieved it upon an eminence apart from less daring minds. It was Loyola’s way of facing human nature and forcing it to concede the supreme self- devotion of his followers. The world looks with amazement upon such actions, but when it sees them, it yields a kind of stupefied allegiance to those who have thus rushed beyond the bounds. And to a close analysis there is as much concealed spiritual pride about
- 36. 306 this nastiness as there is an unnecessary shock given to the sense of decency. Thus, as Mozoomdar says, in his Oriental Christ, “Instead of abasing self, in many cases it serves the opposite end.” It “imposes a sort of indebtedness upon Heaven” (p. 66). Yet the poor wretch who felt those lips upon his awful wound could not but worship the frightful hero who plunged into such nauseous contact with his loathsomeness. Yes, this was and is the power of it all. It was and it is the key-note of much that is potent with the world. When Victor Hugo pictures Jean Valjean in the toils of the Thenardiers laying that white, hot, hissing bar of iron upon his arm and calmly standing before them while they shrink—ogres as they are—from the stench and the sight, he merely uses this same element. Whatever, in short, among us brings out the old savage nature; whatever plunges outside of the conventionalities, the proprieties, or even the common decencies of life; whatever defies the lightning, or dares the volcano, or tramples upon the coiled serpent, that is the thing which controls the world. It is worthy of note that this is not a Christian but a Jesuit act. It is born of that exaggerated sentimentalism which chooses to go beyond Christ and His apostles in its fallacious abnegation of self. But wherever such acts are performed they rank as the marks of saintship and as the stigmata of a crucifixion which proudly places itself on the same Golgotha with another and nobler cross. The records, not merely of Xavier’s life, but of the lives of the saints, swarm with these creeping, slimy frogs of Egypt, raised up by enchanters of the human mind to make Pharaoh believe them to be equal to a far higher Providence. And if we say little in these pages about such strange developments and morbid growths of piety, it need not be forgotten that they existed, and that they have been fostered and encouraged by the Roman Church. The Breviary, for instance, commends a roll of self-flagellators who used the whip upon their naked backs, and Xavier heads the list with his iron flail. Cardinal Damiani, who wrote one of our loveliest hymns, introduced this fashion of scourging in 1056, and the holy nun, St. Theresa,
- 37. 307 after such exercises and an additional repose upon a bed of thorns, was “accustomed to converse with God.” [Aliquando inter spinas volutaret sic Deum alloqui solita.] This topic, with its allied suggestions, is altogether out of our present scope; but in order to see Xavier as he was, we must appreciate to what extent his spirit was subdued before his belief. This was the man, tested and edged and tempered, to whom was now committed the “salvation of the Indies.” It was during the papacy of Paul III., the same Pope who excommunicated Henry VIII. of England. And Xavier, who had practised many austerities both in life and in behavior, was at first sent to Bologna, while Loyola, with Faber and Laynez, went to Rome. It was subsequently at Rome that Xavier had his famous vision, in which he awoke crying, “Yet more, O Lord, yet more!” for he fancied that—as the Apostle Paul once did—he had beheld his future career and was glorying in trials and persecutions. Especially did he often have a dream in which he seemed to be carrying an Indian on his shoulders and toiling with him over the roughest and hardest roads. And when at last Govea, the Rector of the College of Ste. Barbe, happened to be in Rome, Ignatius and his companions were brought by him to the notice of John III. of Portugal, and the king desired to have six of them for use in India. The Pope did not show any special desire to secure their services, and when the question came up he referred it to Ignatius to decide it as he pleased. That sagacious general objected to taking six from ten and leaving only four to the rest of the world, for his ambition now extended to the orb of the earth. He accordingly chose Rodriguez and Bobadilla for India, men who were evidently well selected, for the first became a great propagandist in Portugal, and the other was a decided obstacle to the Reformation in Germany. When Rodriguez, however, fell ill with an intermittent fever Xavier naturally occurred to Loyola as the proper substitute. He therefore commissioned him for the service, and the worn and wasted ascetic patched up his old coat, said farewell to his friends, and having craved the Pope’s blessing, set off from Rome with the Portuguese Ambassador, Mascarenhas, on March 16th, 1540. He
- 38. 308 started in such poverty that Loyola took his own waistcoat and put it upon him, and he left behind him a written paper of consecration to the society, expressing in it his desire that Loyola should be its head, with Faber as alternate, and in which he took the vows of poverty, chastity, and obedience to the order under whose auspices he was going forth. At the Portuguese Court in Lisbon, both Xavier and his companion were diligent in their religious work. The morals of the capital were quite reformed, and when it came time for the ships to sail to the East the king would only spare Xavier and detained Rodriguez, by the advice of Loyola, further to improve the affairs at home. Xavier now sailed as Nuncio with papal commendation and with a poverty of outfit which had its due effect upon his companions on board the ship. The vessel itself was one of those great galleons of Spanish or Portuguese origin, carrying often a thousand persons, and having from four to seven decks. They were huge, unwieldy constructions and were generally freighted with large amounts of rich merchandise. The course was that pursued by Vasco da Gama—around the Cape of Good Hope and into the Indian Ocean —and the voyage often lasted beyond eight months. It is quaintly related of travellers by these precarious sea-paths that they used to take their shrouds and winding-sheets with them in case they died by the way. The company on shipboard was as bad as the provisions, which were often execrable. The peninsular sailors never had the art either of discipline or of storing a ship and supplying what was needful for a voyage, as the English sea-kings had it. Hence their vessels were great floating caravansaries of human beings, full of the scum and offscouring of society—with lords and ladies on the quarter-deck, and robbers and murderers, harlots and gamblers down below. The crew was as prompt as that of Jonah’s ship to cry upon their gods whenever the wind blew. Such inventions as the ship’s pump, the chain-cable, and the bowsprit were not known to them. And when
- 39. 309 we see Sir Richard Grenville in the little Revenge fighting fifteen great Dons for as many hours, or Sir John Hawkins beating his way out of the harbor of Vera Cruz when the Jesus of Lubec was lost by Spanish treachery, we see how utterly cumbrous and awkward these galleons were when compared with English vessels. Sickness also, in the form of fevers and scurvy, was very frequent. And there was such laxity of discipline that a six months’ voyage generally turned the great hulk into a hell of misery and riot. Here, therefore, Xavier was in his element. He slept on the deck; he begged his own bread, and the delicacies pressed upon him by the captain he divided among the neediest of the poor sufferers; he invented games to amuse those who were inclined toward amusement; and by degrees he commingled his sympathy and friendly offices with the necessities of the crew and passengers until they called him the “holy father.” He constantly preached, taught, and labored in this manner until he finally succumbed to an epidemic fever which broke out when they were not far from Mozambique. Here he was landed and for a time was in hospital, at length completing his voyage to India in a different ship from that in which he had first embarked. Scattered through his story, both then and afterward, we have accounts of various miracles, of his exhibition of a spirit of prophecy, and eventually of his raising the dead. These demand a moment’s consideration. He is said, for instance, to have predicted the loss of the San Jago, in which he sailed from Portugal and which was wrecked after he left her. He did the same with one or two other vessels and assured several persons of their own impending death or misfortune. Sometimes he was observed to speak as though he were holding conversation with unseen companions, and he was apparently conscious of events which were afterward found to have occurred at the very time in distant places. There is also a series of phenomena connected with the “gift of tongues” in his case, by which this power appears to have been intermittent, or at least dependent to a great degree upon a remarkable intensity of
- 40. 310 scholarship and keenness of analysis combined with a powerful memory. It is not claimed that he exercised this gift in such a manner as “to converse in a foreign tongue the moment he landed in this foreign country.” And then there is a further class of remarkable experiences connected with fevers and diseases and the raising of the dead. Of these latter miracles it may be well to treat first. He is said to have raised up Anthony Miranda, an Indian, who had been bitten by a cobra; to have restored four dead persons at Travancore; to have resuscitated a young girl in Japan and a child in Malacca, and to have actually brought to the ship, alive and well, a lad who had fallen overboard and been apparently lost. These incidents are related with great gravity by the biographers and are accepted by the faithful as being strictly true. To impugn them is as if one impugned the Scriptures. Nevertheless there is an opening for scepticism in sundry cases, and it may be that we shall do well to agree with the saint’s own statement made to Doctor Diego Borba. “Ah, my Jesus!” he answered, “can it be said that such a wretch as I have been able to raise the dead? Surely, my dear Diego, you have not believed such folly? They brought a young man to me whom they supposed to be dead; I commanded him to arise, and the common people, who make a miracle of everything, gave out the report that a dead man had been raised to life.” For the rest, we may well believe that the same exaggeration and lack of scientific attention to details have accompanied the various accounts, in some such manner as appears in the little sketch of his personal characteristics which a young Coquimban named Vaz has given to us. This enthusiastic admirer describes his going afoot with a patched and faded garment and an old black cloth hat. He took nothing from the rich or great unless he applied it to the uses of the poor. He spoke languages fluently without having learned them, and the crowds which flocked to hear him often amounted to five or six thousand persons. He celebrated Mass in the open air and preached from the branches of a tree when he had no other pulpit. But of this healing of the sick and raising of the dead we are not offered any
- 41. 311 better testimonials than the “Acts of his Canonization.” Moreover, in a manner quite contrary to the experiences recorded in the Gospels, these various miracles seem to be looked upon as the decisive stroke of Christian policy. Upon their occurrence tribes and kingdoms bow before the truth—a thing which did not happen at the tomb of Lazarus, or before the walls of Nain, or within the house of Jairus. In those cases the evangelists are content to tell us that the influence was limited and confined to a very moderate area. Yet when we come to the cures of sick people, to the singular predictions, and to the exalted condition into which Xavier must often have been lifted, we must allow to the man a very high degree of mystical and mesmeric and even clairvoyant power. We are wise enough nowadays to observe the influence of a devoted personality, as when Florence Nightingale traverses the hospital wards at Scutari, or David Livingstone moves through savage tribes, to his dying hour at Lake Lincoln. And when profound Church historians will not altogether discredit the miracles of the Nicene Age which Ambrose and Augustine relate, it causes us to be charitable even toward the miracles of Bernard of Clairvaux, who recorded at large his own sense of uneasiness respecting his power of curing the sick. But it somewhat relieves the mind when the very chapters which relate these experiences of St. Francis Xavier, mention also that a crab came out of the sea and brought him his lost crucifix, and that after he had lived in a certain house two children and a woman fell out of the window at different times and received not so much as a single bruise, though they dropped from an immense height upon the sea-wall. The credulity which includes such palpable absurdities must surely have exposed itself to misstatements and exaggerations in other directions. It is far pleasanter for us to follow Xavier from his arrival at Goa, May 6th, 1542, to the fisheries of Cape Comorin; thence to Malacca, and so to the Banda Islands, Amboyna, and the Moluccas in 1546; again to Malacca in 1547; to Ceylon and back to Goa in 1548, and finally to Japan. In 1551 he planned a visit to China, but was
- 42. 312 disappointed, and at the moment when he was hoping to accomplish a great purpose he died on the island of San Chan, December 22d, 1552, at the early age of forty-six years. Closely studying himself and his methods we find him greatly and always devout, his breviary, however, being his Bible. He prayed much and labored incessantly. His charity to small and great was untiring. He would go through the streets ringing a little bell and calling people to come to religious worship, being frequently attended by a throng of children who seem to have loved him and been beloved by him. He had noble and sweet and modest traits in his character. But we often notice the reliance he places on baptism —sometimes conferring this rite until his arm dropped from weariness. And we observe how much of the wisdom of the serpent can be discerned in his ways with the people whom he desired to secure. The indefatigable exertions of Xavier are above all praise. He never appears to have slackened in his zeal, nor does he ever show hesitation, doubt, or uncertainty of any kind. On one occasion when roused by a great crisis he displayed a military authority worthy of Loyola himself. He stood once in front of an invading host of Badages and forbade them to attack the Paravans, shouting to them, “In the name of the living God I command you to return whence you came.” No wonder that the semi-barbarous people were affected by this fearless and singular presence, and spoke of Xavier as a person of gigantic stature dressed in black and whose flashing eyes dazzled and daunted them. But upon other occasions he was gentle and amenable to every agreeable trait in his companions. He could even take the cards from a broken gamester, shuffle them to give him good fortune, and send him back to try his luck with fifty reals borrowed from another passenger. The man’s success is thereupon made a basis for his penitence. And so with the wicked cavalier of Meliapore, whose friendship he gained by being unconscious of his vices until
- 43. the time for exhortation arrived. In these and similar instances we cannot fail to observe a thorough knowledge of human nature, and a Jesuit’s keen power of using it for his own purposes. He was not always prospered in his enterprises. Once at least he literally shook off the dust from his shoes against an offending tribe. At another time he was wounded by an arrow. But, as a rule, he had a complete moral victory in whatever he undertook. In one of his letters he speaks of the people being maliciously disposed and ready to poison both food and drink. But he will take no antidotes with him, and is determined to avoid all human remedies whatsoever. It is in such superb examples of his absolute trust in God that he presents to us the really grand side of his character. He did not know what fear was, and as for death, he was too familiar with daily dying to be concerned at it. His personal faith was such as to beget faith in others, as when an earthquake interrupted his preaching upon St. Michael’s Day, and he announced that the archangel was then driving the devils of that unhappy country back to the pit. This was said so earnestly as to produce a profound conviction of its truth and to remove all alarm from his audience. But when we are asked to believe that the two Pereiras ever beheld him elevated from the earth and actually transfigured, or when it is stated that he lifted a great beam as though it had been a lath, we must be excused for being doubtful of the statement. There is nothing more destructive of religion than superstition, and nothing which kills faith like credulity. Xavier, with all his false notions, was a most sincere and even majestic figure—a hero of the faith, who shows us the power of a thoroughly devoted spirit unencumbered by any earthly tie and unobstructed by any earthly want. The entire self-immolation of this career constitutes its amazing power. It is the missionary spirit carried to its loftiest height. Perhaps one of his most ingenious ways to secure the good-will of his companions was by endeavoring to excite their benevolence. He would encourage them to little acts of kindness and would repay
- 44. 313 these by similar favors and services. Particularly he used persuasion rather than denunciation, and personal efforts rather than general harangues. He was “all things to all men,” going “privately to those of reputation,” as Paul, his great model, was wont to do. He once wrote: “It is better to do a little with peace than a great deal with turbulence and scandal.” On April 14th, 1552, he set sail from Goa for Malacca where a pestilence was raging. This delayed him awhile from China, and he was held back still longer by the envious quarrellings of those who aspired to the honor of attending him on his voyage. Xavier was reduced to the necessity of producing the papal authority which constituted him Nuncio, and of threatening with excommunication Don Alvaro Ataïde, the most troublesome person. In addition to this difficulty he found himself insulted and reviled in the open street, but accepted everything with meekness and patience; which, however, did not prevent his finally excommunicating Ataïde in the regular form. The vessel on which he embarked was manned mostly by those in the pay of Ataïde, but he did not shrink from the voyage. The voyage itself is decorated with many legends, as might be expected. The saint is reported to have changed salt water into fresh; to have rescued a child from death in a miraculous manner, and to have become suddenly so much taller and larger than those about him as to have been compelled to lower his arms when he baptized the converts. They sailed from Chinchoo to San Chan, an island in which the Portuguese had some trading privileges. It was here that Xavier uttered a prediction which may serve to explain other singular occurrences. He would seem to have possessed more than an ordinary amount of medical skill in diagnosis, and looking earnestly upon an old friend named Vellio, he bade him prepare for death whenever the wine he drank tasted bitter. This might easily be from either of two causes—poison, or a disorganized state of the system. And it is recorded that the result fulfilled the prophecy. Nor is there much doubt that Vellio’s entire faith in the prediction helped on his death.
- 45. 314 From San Chan Xavier now proposed to cross to China. He arranged to be smuggled thither in a small boat, but the residents of San Chan, English as well as Portuguese, became alarmed at the consequences which they foresaw from this desperate scheme of intrusion into the forbidden empire. And to crown all his woes he fell sick with a fever, from which, however, he convalesced in a fortnight. He was now more anxious than ever to go on with his project. But all the Portuguese ships had sailed back again except the Santa Cruz, on which he had arrived. And now he was truly deserted and neglected. He had scarcely the bare necessaries of life, sometimes being deprived entirely of food. The sailors were mostly in Ataïde’s pay and inimical to his purpose. At length he became convinced that he would himself soon die, and so would often walk in meditation and prayer by the seashore gazing toward the prohibited coast. At this time the young Chinese Anthony was his only hope as an interpreter; and he was now deprived of the services of the merchant and his son who had agreed to row him over to Canton. They had deserted him, and only Anthony and one more young lad remained true to the dying missionary. On November 20th the fever again seized him after he had celebrated Mass. He was taken to a floating hospital, but being disturbed by its motion he begged to be landed. This was done and he was left upon the beach in the bleak wind. A poor Portuguese named George Alvarez then took pity on him and removed him to his own hut of boughs and straw. Rude medical care was given him, but on December 2d, about two o’clock in the afternoon, he had reached the limit of his life. His latest words were, In te, Domine, speravi—non confundar in aeternum—O Lord, I have trusted in Thee, I shall never be confounded, world without end. Thus died Francis Xavier, for ten years and seven months a missionary in the most dangerous and deadly regions of the earth. At the date of his death he was of full and robust figure in spite of his privations, with eyes of a bluish-gray, and hair that had changed
- 46. 315 its dark chestnut color somewhat through his toils and sufferings. His forehead was broad, his nose good, and his expression pleasant and affable. His beard, like his hair, was thick, and his temperament was nearly a pure sanguine. They buried him first at San Chan, then removed him to Goa, where in solemn procession they conducted his mortal body to its final rest. But his right arm was taken off and it is to be observed that “the saint seems not to have been pleased at the amputation of his arm,” which, however, did not prevent the Jesuit, General Claude Acquaviva, from insisting upon the mutilation. Down to the present time his memory has received many honors. Churches have been erected, prayers have been offered, and much religious worship has been transacted in his name. But to us who are looking upon him from another angle altogether, there are apparent in him a piety, a zeal, a courage, and a “hot-hearted prudence” (to quote F. W. Faber’s words) which arouse our admiration. And in the two hymns which bear his name we are able to discover that fine attar which is the precious residuum of many crushed and fragrant aspirations, which grew above the thorns of sharp trial and were strewn at last upon the wind-swept beach of that poor Pisgah island from which he truly beheld the distant Land. O DEUS, EGO AMO TE. O Lord, I love thee, for of old Thy love hath reached to me. Lo, I would lay my freedom by And freely follow thee! Let memory never have a thought Thy glory cannot claim, Nor let the mind be wise at all Unless she seek thy name.
- 47. 316 For nothing further do I wish Except as thou dost will; What things thy gift allows as mine My gift shall give thee still. Receive what I have had from thee And guide me in thy way, And govern as thou knowest best, Who lovest me each day. Give unto me thy love alone, That I may love thee too, For other things are dreams; but this Embraceth all things true.
- 48. CHAPTER XXVIII. THE HYMN-WRITERS OF THE BREVIARY. There are three principal liturgical books in use in the Roman Catholic Church. Originally there were two: the Ritual, which contained all the sacramental offices, and the Breviary, which contained the rest. But for convenience the eucharistic office in its various forms now has a book to itself called the Missal, and the other six sacraments recognized in the Church of Rome make up the Ritual. It is with the Breviary, however, that hymnology is especially concerned, as it is in it that the hymns of the Church are mostly to be found, while the sequences belong to the Missal. It contains the prayers said in the Church’s behalf every day at the canonical hours by the priests and the members of the religious orders. Originally there were only three of these canonical hours, and they were based on Old Testament usage. These were at the third, sixth, and ninth hour of the Scriptures (nine o’clock, noon, and three in the afternoon), and in the Western Church are called Tierce, Sext, and Nones, for that reason. The number afterward was increased to five and then to seven. To these three day hours were added three night hours, with two at the transition from night to day (Prime), and from day to night (Vespers). But to get up thrice in the night was too much for even monastic discipline, so they said two night services together at midnight, and then they slept till dawn. As this daily service differs in its contents according to the seasons of the Church year, and also is adapted to the commemoration of the saints of the Calendar, the Breviary is the most voluminous prayer-book known to Christendom. It generally is published in four substantial volumes, one each for the four natural seasons. It is used in such public
- 49. 317 services as are not accompanied by a celebration of any sacrament and in the choir service of the religious houses. In theory, however, the Church is present even at the solitary recitation of the hours by a secular priest; and when two say them in company they must say them aloud. Hymns were not in the services of the Breviary from the beginning. As late as the sixth century there was a controversy as to admitting anything but the words of Scripture to be sung. We find a Gallic synod sanctioning their use, and a Spanish synod taking common ground with our Psalm-singing Presbyterians. But in the next century even Spain, through the Council of Toledo (A.D. 633), appeals to early precedent in behalf of hymns, and decides that if people may use uninspired words in prayer, they may do the same in their praises—Sicut ergo orationes, ita et hymnos in laudem Dei compositos nullus vestrum ulterius improbet—which went to the core of the question and silenced the exclusive Psalm-singers. Twenty years later another Council of Toledo required of candidates for orders that they should know both the Psalter and the hymns by heart. Yet in the Roman Breviary no hymns were introduced before the thirteenth century, when Haymo, the General of the Franciscan Order, reformed it in 1244 with the sanction of Gregory IX. and Nicholas III. In the view of Roman Catholic liturgists, the Psalms set forth the praise of God in general, while hymns are written and used with reference to some single mystery of the faith, or the commemoration of some saint. This harmonizes with their use in the Breviary, and their division into hymns de tempore for the festivals of the Church year, or the days of the week, or the hours of the day; and hymns de sanctis for the days of commemoration in the Church Calendar. Even when the same hymn is used on a series of days, its conclusion is altered to give it a special adaptation to each of these days. This classification, of course, does not describe the whole body of the Latin hymns. Some few even of those in the Breviary, as, for instance, the Te Deum, have to be classed as psalms, and are called
- 50. 318 Canticles (Cantica); and many outside it will not fit into any such definition of what a hymn is. But it illustrates the general character and purpose of the hymns of the Roman and other breviaries, as designed for a special temporal or personal application by way of supplement to the Psalter. At present the Roman Breviary, prepared with the sanction of the Council of Trent, has driven nearly all the others out of use. But at the era of the Reformation there was a great number of breviaries, every diocese and religious order having a right to its own. Panzer enumerates no less than seventy-one which were printed before 1536, some of them in several editions. [18] Even now the Roman Breviary is supplemented by special services in honor of the saints of each order or country, and by services of a more general kind which are peculiar to some localities. But in Luther’s time the endless variety in breviaries and missals formed a striking feature of the confusion which to his mind characterized the Church of Rome. With the development of a more fastidious taste, through the study of the Latin classics as literary models, there arose in the sixteenth century, and even before the Reformation, a demand for a reformation of the Breviary. Besides its defects of form, such as violations of Latin grammar, the constant use of terms which grated on the ears of the humanists, and the use of hymns in which rhyme rather added to the offence of want of correct metre, the contents of the Breviary were found faulty by a critical age. The selections from the Fathers to be read by way of homily were in some cases from spurious works; and the narratives of saints’ lives for the days dedicated to them were not always edifying, and in some cases palpably untrue. It became a proverbial saying that a person lied like the second nocturn office of the Breviary, that being the service in which these legends are found. But the badness of the Latin and the metrical faults of the hymns counted for quite as much with the critics of that day. We hear of a cardinal warning a young cleric not
- 51. 319 to be too constant in reading his Breviary, if he wished to preserve his ear for correct Latinity. As might have been expected, it was the elegant Medicean Pope Leo X. who first put his hand to the work of reform. He selected for this purpose Zacharia Ferreri, Bishop of Guarda-Alfieri, a man of fine Latin scholarship and some ability as a poet. By 1525 Ferreri had the hymns for a new Breviary ready, and published them with the promise of the Breviary itself on the title-page. [19] Clement VII., also of the house of Medici, was Pope when the book appeared, and he authorized the substitution of these new hymns for the old, but did not command this. The book is furnished with an introduction by Marino Becichemi, a forgotten humanist, who was then professor of eloquence at Padua. It is worth quoting as exhibiting the attitude of the Renaissance to the earlier Christian literature. He praises Ferreri as a shining light in every kind of science, human and divine, prosaic and poetical. He cannot say too much of the beauty of his style, its gravity and dignity, its purity, its spontaneity and freedom from artificiality. “That his hymns and odes, beyond all doubt, will secure him immortality, I need not conceal. Certainly I have read nothing in Christian poets sweeter, purer, terser, or brighter. How brief and how copious, each in its place—how polished! Everywhere the stream flows in full channel with that antique Roman mode of speech, except where of full purpose it turns in another direction.” That means how Ciceronian Ferreri’s speech, except where he remembers that he is a Christian poet and bishop writing for Christian worshippers. “More than once have I exhorted him that it belonged to the duty and dignity of his episcopal (pontificii) office to make public these Church hymns.” “You know, my reader, what hymns they sing everywhere in the temples, that they are almost all faulty, silly, full of barbarism, and composed without reference to the number of feet or the quantity of
- 52. 320 the syllables, so as to excite educated persons to laughter, and to bring priests, if they are men of letters, to despise the services of the Church. I say men of letters. As for those who are not, and who are the gluttons of the Roman curia, or who have no wisdom, it is enough for them to stand like dragons close by the sacred ark, or to drift about like the clouds, to live like idle bellies, given over to the pursuit of sleep, good living, sensual pleasures, and to gather up the money by which they make themselves hucksters in religion and plunderers of the Christian people and practice their deceits upon both gods and men equally, until the vine of the Lord degenerates into a wild plant.” The Italianized Greek would see no difference between a Tetzel and a Ferreri. But there still were sincerely good people who relished the old hymns better than the polished paganism of the Bishop of Guarda-Alfieri. Ferreri’s hymns struck no root in spite of the favor of two Medicean popes. They seem never to have reached a second edition. Their frankly pagan vocabulary for the expression of Christian ideas seems to have been too much for even the humanists. Bishop Ferreri does not seem to have lived to prepare his shorter and easier Breviary after the same elegant but unsuitable fashion as his hymns. So Clement VII. put the preparation of a new Breviary into the hands of another and a better man, Cardinal Francesco de Quiñonez. He was a Spanish Franciscan, had been general of his order, and was made Cardinal by Clement in acknowledgment of diplomatic services. He enjoyed the confidence of the Emperor Charles V., and used it to rescue the Pope from his detention in the Castle of San Angelo, when he was besieged there after the taking of Rome by the Imperial troops in 1529. This is hardly the kind of record which would lead us to look for a reformer under the red hat of our cardinal. But, so far as the Breviary was concerned, he proved himself too rigorous a reformer, if anything. His work was governed by two leading principles. The first was to simplify the services by dropping out those parts which had been added last. The second
- 53. 321 was to use the space thus obtained to insert ampler Scripture lessons and more Psalms, so that, as in earlier times, the Bible might be read through once a year and the Psalter once a week. It is this last feature which has elicited the praise of Protestant liturgists, and it is known that the Breviary of Quiñonez furnished the basis for the services of the Anglican Book of Common Prayer, excepting, of course, the Communion Service. But unfortunately hymnologists are not able to join in this praise. To get the Psalms said or sung through once a week, he dealt nearly as ruthlessly with the hymns as if he were a Seceder. His Breviary appeared in 1535, [20] and for thirty-three years its use was permitted to ecclesiastics in their private recitation of the hours. It appeared in a large number of editions in different parts of Europe, so that its use must have been extensive. But it did not pass unchallenged. The doctors of the Sorbonne at Paris hurried into the arena with their condemnation of it before the ink was fully dry on the first copies. They declared it a thing unheard of to introduce into Church use a book which was the production of a single author, and he—as they wrongly alleged—not even a member of any religious order. Furthermore, he had so shortened and eviscerated the legends for the saints’ days, besides omitting many, that nobody could tell what virtues and what miracles entitled them to commemoration. Above all he had omitted Peter Damiani’s Little Office of the Blessed Virgin! Much better founded was the objection to the omission of parts long established in use, such as the antiphons and many of the hymns. Here we must side with the Sorbonne against Quiñonez. It was not until 1568 that the present Roman Breviary appeared. When the Council of Trent met in its final session in 1562, the first drafts of a reformed Breviary and Missal were transmitted to the Fathers by Pius IV.; but they were too busy with questions of discipline to do more than return these with their approbation. The work was published by Pius V. in July, 1568, and its use was made
- 54. 322 obligatory upon all dioceses which had not had a Breviary of their own in use for two hundred years previously. This is in substance the Breviary now in use throughout the Roman Catholic Church. It underwent, however, two further revisions. That under Clement VIII., finished in 1602, was by a commission in which Cardinals Bellarmine, Baronius, and Silvius Antonianus were members. That under Urban VIII., completed in 1631, concerns us more directly, and especially the part of it which was effected by three learned Jesuits: Famiano Strada, Hieronimo Petrucci, and Tarquinio Galucci, who had in their hands the revision of the hymns. The three revisers, all of them poets of some distinction, and the first famous for his history of the wars in the Low Countries, had to steer a middle course in the matter of revision. None of them were radical humanists after the fashion of Zacharia Ferreri; that fashion, indeed, had gone out with the rise of the counter-reformation and of the great order to which they belonged. Yet in the matter of “metre and Latinity,” of which Ferreri boasted on his title page a hundred years before, the revival of classical scholarship had established a standard to which the old hymns even of the Ambrosian period did not conform. The revisers profess their anxiety to make as few changes as possible; but Pope Urban, in his bull Psalmodiam sanctam prefixed to the book, announces that all the hymns—except the very few which made no pretension to metrical form—had been conformed to the laws of prosody and of the Latin tongue, those which could not be amended in any milder way being rewritten throughout. Bartolomeo Gavanti, a member of the Commission of Revision, but laboring in another department, tells us that more than nine hundred alterations were made for the sake of correct metre, with the result of changing the first lines of more than thirty of the ninety-six hymns the Breviary then contained; that the three by Aquinas on the sacrament, the Ave Maris stella, the Custodes hominum, and a very few others, were left as they were.
- 55. This, then, is the genesis of the class of hymns designated in the collections as traceable no farther back than the Roman Breviary. Some of them are original, being the work of Silvius Antonianus, Bellarmine, or Urban VIII. himself, or of authors of that age whose authorship has not been traced. But the greater part are recasts of ancient hymns to meet the demands of the humanist standards of metre and Latinity. It is not easy to give a merely English reader any adequate idea of the sort of changes by which Strada and his associates adapted the old hymns to modern use. But for those who can read Latin some specimens are worth giving. Take first the great sacramental hymn of the eighth or ninth century: Ad coenam Agni providi Et stolis albis candidi, Post transitum maris Rubri Christo canamus principi, Cujus corpus sanctissimum In ara crucis torridum, Cruore ejus roseo Gustando vivimus Deo Protecti paschae vespero A devastante angelo Erepti de durissimo Pharaonis imperio. Jam pascha nostrum Christus est Qui immolatus agnus est, Sinceritatis azyma Caro ejus oblata est. O vera digna hostia Per quam fracta sunt tartara Redempta plebs captivata,
- 56. 323 Reddita vitae praemia Cum surgit Christus tumulo Victor redit de barathro, Tyrannum trudens vinculo, Et reserans paradisum Quaesumus, auctor omnium In hoc paschali gaudio: Ab omni mortis impetu Tuum defende populum. Ad regias Agni dapes Stolis amicti candidis Post transitum maris Rubri Christo canamus principi:
- 57. Divina cujus charitas Sacrum propinat sanguinem, Almique membra corporis Amor sacerdos immolat Sparsum cruorem postibus Vastator horret angelus: Fugitque divisum mare Merguntur hostes fluctibus. Jam Pascha nostrum Christus est Paschalis idem victima, Et pura puris mentibus Sinceritatis azyma O vera coeli victima Subjecta cui sunt tartara, Soluta mortis vincula, Recepta vitae praemia Victor subactis inferis Trophaea Christus explicat, Coeloque aperto, subditum Regem tenebrarum trahit. Ut sis perenne mentibus Paschale, Jesu, gaudium: A morte dira criminum Vitae renatos libera. Now it is impossible to deny to the revised version merits of its own. Not only does it use the Latin words which classic usage requires— as dapes in poetry for coena, recepta for reddita, inferis for barathro —but it brings into clearer view the facts of the Old Testament story which the hymn treats as typical of the Christian passover. The (imperfect) rhyme of the original is everywhere sacrificed to the
- 58. 324 demands of metre, which probably is no loss. But the gain is not in simplicity, vigor, and freshness. In these the old hymn is much superior. The last verse but one, for instance, presents in the old hymn a distinct and living picture—the picture Luther tells us he delighted in when a boy chorister singing the Easter songs of the Church. But in the recast the vividness is blurred, and classic reminiscence takes the place of the simple and direct speech the early Church made for itself out of the Latin tongue. Take again the first part of the dedication hymn, of which Angulare fundamentum is the conclusion: Urbs beata Hierusalem Dicta pacis visio Quae construitur in coelis Vivis ex lapidibus Et angelis coronata Ut sponsata comite Nova veniens e coelo Nuptiali thalamo Praeparata, ut sponsata Copulatur domino, Plateae et muri ejus Ex auro purissimo Portae nitent margaritis Adytis patentibus, Et virtute meritorum Illuc introducitur Omnis, qui pro Christi nomine Hoc in mundo premitur Tunsionibus, pressuris Expoliti lapides Suis coaptantur locis
- 59. Per manum artificis, Disponuntur permansuri Sacris aedificiis. Coelestis urbs Jerusalem Beata pacis visio Quae celsa de viventibus Saxis ad astra tolleris, Sponsaeque ritu cingeris Mille angelorum millibus. O sorte nupta prospera, Dotata Patris gloria, Respersa Sponsi gratia Regina formosissima, Christo jugata principi Coelo corusca civitas. Hic margaritis emicant Patentque cunctis ostia, Virtute namque praevia Mortalis illuc ducitur Amore Christi percitus Tormenta quisquis sustinent. Scalpri salubris ictibus Et tunsione plurima, Fabri polita malleo Hanc saxa molem construunt, Aptisque juncta nexibus Locantur in fastidia. Daniel in his first volume prints fifty-five of these recasts in parallel columns with the originals, and to that we will refer our readers for further specimens. It is gratifying to know that not all the
- 60. 325 scholarship of that age was insensible to the qualities which the revisers sacrificed. Henry Valesius, although only a layman and a lover of good Latin—as his versions of the historians of the early Church show—uttered a fierce but ineffectual protest in favor of the early and mediaeval hymns. And the Marquis of Bute, a convert to Catholicism, who published an English translation of the Breviary in 1879, says that the revisers of 1602 “with deplorable taste made a series of changes in the texts of the hymns, which has been disastrous both to the literary merit and the historical interest of the poems.” He hopes for a further revision which shall undo this mischief, but in other respects return to the type furnished by the Breviary of Quiñonez. The translations from the hymns of the Roman Breviary have been very abundant. Those by Protestants have been due to the fact that the texts even of ancient hymns were so much more accessible in their Breviary version than in their original form. Among Roman Catholics, of course, other considerations have weight; and in Mr. Edward Caswall’s Lyra Catholica and Mr. Orby Shipley’s Annus Sanctus will be found some very admirable versions. The latter book is an anthology from the Roman Catholic translators from John Dryden to John Henry Newman. From the Breviary text Mr. Duffield has made the following translations of two hymns by Gregory the Great: JAM LUCIS ORTO SIDERE. Now with the risen star of dawn, To God as suppliants we pray, That he may keep us free from harm, And guide us through an active day. May he, restraining, guard the tongue, Lest it be found to strive and cry, And, lest it drink in vanities,
- 61. 326 May he protect the wayward eye. Let all our inmost thoughts be pure, And heedlessness of heart be gone; Let self-denying drink and food Hold pride and flesh securely down, That when the day at length is past, And night in turn has come to men, Through abstinence from earth, we may Give thee the only glory then. To God the Father be the praise, And to his sole-begotten Son, And to the Holy Paraclete, Now and until all time be done. ECCE JAM NOCTIS TENUATUR UMBRA. Lo, now, the shadows of the night are breaking, While in the east the rising daylight brightens, Therefore with praises will we all adore thee, Lord God Almighty! How doth our God, commiserating mortals, Drive away sorrow, offering them safety, Since he shall give us, through paternal kindness, Rule in the heavens! This let the blessed Deity afford us, Father and Son and equal Holy Spirit, Whose through the earth be glory in all places Ever resounding.
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