Arne Beurling

A Swedish documentary about cryptanalysis during WWII:

http://www.youtube.com/watch?v=QrWup70DdTc

At around 8:30 one can hear eminent mathematician Lennart Carleson, winner of the Abel Prize, say the following about Beurling:

"He had a relation to mathematics that was a bit mystical. It was as if he had a direct line to God, in some way, with information and knowledge that we didn't have."

Heisenberg's platonism

W. Heisenberg was one of the great physicists of the 20th century, and helped shape modern quantum mechanics. Around two thousand years after Plato and Aristotle, and even before the more recent developments of mathematical physics such as string theory, a leader of modern science has this to say:

I think that modern physics has definitely decided in favor of Plato. In fact the smallest units of matter are not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language.

Hardy's platonism

G. H. Hardy was a famous number theorist in Cambridge, and the mentor of Srinivasa Ramanujan. He wrote a short book called A Mathematician's Apology, explaining his view of mathematics. It is a rather cynical book, devoid of any 'romanticism' about the mathematical profession. Hardy was sceptical of religion, and probably of spirituality in general. Nevertheless, he was a fully-fledged Platonist:

I believe that mathematical reality lies outside of us, and that our function is to discover, or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations' are simply notes on our observations.

Poem

See skulking Truth to her old cavern fled
Mountains of Casuistry heap'd o'er her head!
Philosophy, that lean'd on Heav'n before
Shrinks to her second cause, and is no more.
Physic of Metaphysic begs defence
And Metaphysic calls for aid on Sense!
See Mystery to Mathematics fly!

Alexander Pope, (1688-1744)

Manin's platonism

In the latest issue of the Notices of the American Mathematical Society, one can find an interview with eminent mathematician Yuri Manin, who is very highly regarded in the fields of algebraic geometry, number theory, and mathematical physics. He is also a mathematician who possesses a great breadth of cultural and historical knowledge.
Philosophers sometimes criticize mathematicians who express platonist leanings for being philosophically naive, or for not being aware of the latest developments in thought or empirical science. Examples can be found in earlier posts on this blog, where young Fields medalists were rebuked for expressing platonist opinions. In this way the huge number of platonist mathematicians are arrogantly dismissed as not having reflected on their subject maturely and according to current philosophical standards. This is definitely something that Manin cannot be accused of.

The interview reveals that Manin is not afraid of using the 'P'-word (my italics):

What has changed in pure mathematics?
The unique possibility of doing large-scale physi-
cal experiments in mental reality arose. We can
try the most improbable things. More exactly, not
the most improbable things, but things that Euler
could do even without a computer. Gauss could
also do them. But now, what Euler and Gauss could
do, any mathematician can do, sitting at his desk.
So if he doesn’t have the imagination to distin-
guish some features of this Platonic reality, he can
experiment. If some bright idea occurs to him
that something is equal to something else, he can
sit and sit and compute a value, a second value,
a third, a millionth. Not only that. People have
now emerged who have mathematical minds, but
are computer oriented. More precisely, these
sorts of people were around earlier, but, with-
out computers, somehow something was missing.
In a sense, Euler was like that, to the extent that
he was just a mathematician—he was much more
than just a mathematician—but Euler the math-
ematician would have taken to computers passion-
ately. And also Ramanujan, a person who didn’t
even really know mathematics. Or, for instance, my
colleague here at the institute, Don Zagier. He has
a natural and great mathematical mind, which is at
the same time ideally suited to work with comput-
ers. Computers help him study this Platonic reality,
and, I might add, quite effectively.

At this stage it is not clear what Manin believes regarding platonism. He refers to the mathematical universe as both a "mental reality" and (twice) as a "Platonic reality". This mystery of Manin's opinions deepens even further a little later on, when he says

I am an emotional Platonist (not a rational one: there are no rational
arguments in favor of Platonism). Somehow or other, for me mathematical research is a discovery, not an invention.
I imagine for myself a great castle, or something like that, and you gradually start seeing its contours through the deep mist, and begin to investigate something. How you formulate what it is you’ve seen depends on your type of thinking and on the scale of what you have seen, and on the social circumstances around you,
and so on.

This is puzzling for several reasons. First of all, to say that there are no rational arguments of platonism is not true in the most reasonable and common interpretations of these words. There are about as many rational arguments for platonism as there are for most other philosophical opinions. What Manin must have had in mind is the fact that platonism cannot be fully described in terms of today's science (which works in the paradigm of space-time and materialism).
Nevertheless, Manin seems to truly believe that mathematical platonism is a fact since he clearly states that he believes mathematics is discovered, not invented. His belief is reaffirmed in his platonist confession:

Gelfand: Are these properties of the problems
themselves, or is it just that no one is actively inves-
tigating them, for some social reason?
Manin: As a Platonist, I know that this is a
property of the problems themselves, but it is a
property that one cannot recognize at the moment
of formulating the problem. It reveals itself in the
process of historical development.
Partly for this reason, I am not partial to prob-
lems. Solving a problem requires the skill of
finding a detail, but you don’t know what it is a
detail of. As a Platonist, I am partial to complete
programs. A program arises when a great math-
ematical mind sees something as a whole, or not
as a whole, but as something more than a single
detail. But it is seen at first only vaguely.

The conclusion is that Manin has deeply held platonistic beliefs, but admits he has no scientific explanation for how the mathematical realm can exist in the way posited by platonism.

Pythagoras

The vision of Pythagoras 2500 years ago that number, shape, and relation underlie reality and pervade human knowledge, is in light of the tremendous success of mathematical methods in science and the digitilization of music, one of the most successful predictions made in human history.

Mathematical aristotelianism

Aristotle's philosophy was a reaction to that of his predecessor and teacher Plato. One particular detail that separates the two philosophers is the question of the forms, or universals. While Plato claimed that the forms are eternal, existing independently of their physical actualisations, Aristotle thought that the forms were inherent in the objects, without any possibility of existing independently of the objects. Consequently, mathematical Platonism sees physical objects as instantiations of general forms, while mathematical Aristotelianism views mathematical forms as attributes of particular concrete objects. The question is, do particular objects (such as physical bodies) require the platonic forms for their existence, or do, as Aristotle claimed, the forms depend on particular objects for their existence?
At least when it comes to mathematics, the Aristotelian view runs into trouble because there are mathematical objects and results which cannot possibly have a physical basis. Let's list some examples:

• Infinite sets. Physics tells us that there are only a finite number of (observable) physical entities. (There may exist infinitely many distinct universes, but if you believe this, then you don't need to be convinced of mathematical Platonism.) Aristotle tried to solve this problem by rejecting the existence of 'actual' infinity, and replacing it by 'potential' infinity. It's not entirely clear whether this distinction makes any sense when analysed closer, but in any case modern mathematics could definitely not work without something like actual infinity. Moreover, it is a fact that modern mathematics has tremendous applications to all aspects of science. Hence mathematics is something that uses concepts without physical ground, and at the same time it is something whose consequences deeply affects physical reality.
• The Banach-Tarski paradox. If this result had a physical grounding, it would most likely have to be its standard interpretation involving two physical objects and decompositions of these. However, the mathematical result then implies something which is physically impossible, namely that one solid ball can be decomposed and the pieces rearranged so that to form two balls, each of the same volume as the first. The Banach-Tarski theorem can therefore not possibly have any physical grounding. Thus we have here another example of a mathematical form which cannot possibly be realised physically.

There are more examples of this kind, but it will not change anything to include these. To these examples the Aristotelian can reply that 'real' mathematics is grounded in physical objects, such as a number of pebbles on the ground, and then mathematicians dream up all sorts of abstractions on top of that, which are are just mental constructions with no connection to the real world. Well, the problem with this view is that these 'dreamt-up' abstractions have applications to the physical world which are just as intimate as the relation between three pebbles and the number three. To take a simple example, consider differential and integral calculus discovered by Newton and Leibniz. Operating with infinite sums of infinitely small quantities led to modern calculus and the continuum, techniques essentially incorporating infinite objects, and without which most of physics and its applications in technology, would not have seen the light of day.