I will attempt to defend the claim that not only do numbers and all of mathematics exist in and of themselves, but that in fact "ideas" have a real, concrete, physical existence within our universe. Just to push the claim to an interesting extreme, I will state that ideas are true, living, conscious and self-aware beings; and that mathematics is but a label for a certain subset or species of these beings. I am well aware of how preposterous this may sound to the uninitiated. In fact, its not a terribly new idea: there's a a popular little sub-branch of cultural anthropology devoted to this and related phenomena: memetics, wherein highly contagious, 'viral' ideas are referred to as memes.
The Reality Club debates a good bit about the brain, the mind and their relationship. Surprisingly, not once do they seem to touch on memory, without which the brain/mind would not be the what it is. Memory is vital to the function of all aspects of the mind/brain. A very important development has been the discovery of writing as a means of enhancing both short term and long term literary, legal, financial, musical and mathematical memory. That is, writing is used not only as a means of communication, but also to give permanence to ideas beyond the frail existence of the gelatinous brain. Ars longa, Vita brevis.
Ideas committed to writing have the power to exist not only for long periods of time (as clearly Ancient Greek writings testify), but also come alive again as the newest generation of minds come into contact with them. Although writings today must rely on the human mind to be called into active existence, it seems natural to assume that writings could have the same effect on any possible future artificial intelligence or any non-terrestrial intelligence that may exist. Indeed, the very definition of intelligence seems to imply that it is a fertile petri dish that allows ideas to thrive and grow.
Are ideas "alive"? The science and science-fiction communities both seem to have a common currency for recognizing "life": The ability to affect its surroundings, the ability to grow and change, the ability to metabolize and utilize energy, the ability to breed, multiply and propagate. A living thing may possibly have a concept of self, and the ability to defend itself from attack and destruction. By virtually any criteria that one may want to apply to distinguish "life" from "non-life", an "idea" passes the test for being a "living thing". Lest I confuse readers from this point onwards, I would like to use the word "meme" interchangeably with the word "idea": The word "idea" not only carries historical baggage that the word "meme" does not, but it can lead to syntactic constructions that are hard to parse.
It is very tempting at this point to leave as an exercise for the reader the test of "memes" for the property of "life". But some exploration is worthwhile. Memes certainly have the ability to affect the physical universe: memes have lead to world wars and to great achievements. Ideas make bulldozers move and incite riots: the pen is mightier than the sword. Ideas come and go: some previously popular ones appear to lie dormant, all but dead as they are replaced by more robust ones in a process of natural selection. After all, the human brain is a scarce resource, and ideas compete for the use of this resource. Some ideas intermingle and cross-pollenate to give rise to new offspring, others do not, and indeed appear hostile to each other: creationism vs. evolution being a famous example. Ideas often have a very clear sense of self: sharp minds continually distinguish "relevant" from "irrelevant" in any train of thought. Some ideas are even self-aware: certainly, the idea that I am writing down right now seems to be aware of its own existence and expression.
As long as an intelligent human race continues to exist, ideas will continue to exist; and although some go dormant as others are born, their seeming existence and power in this universe are not threatened unless the human race itself is. Like the mind, they cannot exist in active form without the brain. Unlike the mind, they can take relatively safe shelter in writing and art, lying dormant until the next intelligence can give them an active expression. And given an appropriately large and well-constructed time-capsule, there is good reason to believe that the ideas that the human race has known could well survive the total destruction of the human race, lying in wait for the next extraterrestrial archeologist to evoke them back into life. With an ample enough context, written ideas can indeed survive the loss of language, as any scholar of dead and archaic languages can attest.
What then, does any of this have to do with the "existence of mathematics"? Clearly, mathematical ideas are indeed "ideas", in all of the sense discussed above. How is it that the Reality Club is so willing to abandon the notion that numbers or mathematics do not have an objective reality? Is there a shared belief that there may exist a higher intelligence that would not ever stumble across mathematics, because their brain is not wired in the same way that our motor cortexes are? That indeed the very notion of an idea is predicated on the particular wiring in our heads? To quote Dehaene: "Yet as a basic category of experience provided by a dedicated brain circuit, number is as undefinable as color, space, movement, happiness, or beauty. " Excuse me? Undefinable to whom? There's that old engineering joke about Zeno's paradox: A man and woman stand on opposite sides of the room. They approach, halfway. They approach, another fourth. An eighth. A sixteenth. Will they ever meet? Dunno, but they'll get close enough for all practical purposes.
Of course there are deep philosophical problems with defining color, space, truth or beauty. This does not prevent the human species from having a 'good enough' working definitions of these terms, culturally biased and relativistic as they may be. There is good reason to believe that modern rational analysis not only leads away from cultural / social / anthropological relativism, but is converging on more philosophically correct absolutist concepts that begin to elevate some of the mushier concepts (e.g. beauty) towards a more 'platonic' existence. The corner that Dehaene seems to be painting himself into seems to be that these concepts would not be transmittable to non-human extra-terrestrials. (He does point out that intelligent animals, our evolutionary forebears, seem to have at least some rudimentary neural circuits for mathematical processing, thereby implying they might also have rudimentary mathematical abilities.) So the question is, 'can we share these concepts with extra-terrestrials'? I dunno, this seems so absurd a question, that I tire in the conversation.
As a parting shot: A common phenomenon among young Art students is 'de Kooning love'. To the unaccustomed eye, the paintings of Wilhelm de Kooning are best described as ugly, misanthropic, jagged, messy splotches of paint that don't deserve to be called 'Art'. Yet virtually all art students go through a phase where they suddenly 'get it', an 'ahah experience' where de Kooning suddenly makes sense, and is no longer ugly, but sublimely beautiful. The recognition is frequently so powerful and overwhelming that the student can't cope: they develop this love for de Kooning's works that is less a love as an embarrassingly naive kind of a 'high-school crush' or 'puppy-love'. Eventually, they get over it, which is not to detract from de Kooning: after all, he is in art history books for a reason. The question I then apply: could an extra-terrestrial stranger, e.g. some intelligent assemblage of magneto-hydrodynamic vortexes, could eventually come about to view Wilhelm de Kooning's paintings as 'beautiful, and sublimely so'? You decide. You might already guess my dogma: 'of course they could'. The Hitch-hiker's Guide to the Universe may be filled with all sorts of cranky and difficult intelligent species, but, in my personal experience, relativistic judgments anchored in cultural mores and fashions are all too easily overturned by a bit of open-mindedness, thought and experimentation. Although we have no absolute, scientific, philosophical framework for judging beauty, professional artists and art historians seem not to have any practical problems voicing absolutist opinions about Art that appear to transcend cultural boundaries. They 'know it when they see it'. I would like to believe that the same mechanisms operate for any intelligence, not just terrestrial, civilizationaly contextual ones.
Hmmm. What is the physical nature of mathematical vision? How can I ascribe any reality for Peano's Axioms, given that they are unprovable assertions believed to lead to only non-contradictory deductions about numbers? How can I anchor Peano's axioms more thoroughly into this world, the concrete world, since that is what ultimately a true Platonist must do?
Penrose points out that Godel's and Tuning's theorems about the incompleteness of axiomatic systems depends on the reader's perception that there is a true statement that is not provable. If its not provable, how is it that we 'know' that its true? What is knowledge? GJ Chaitin asserts that Godel's theorem is just the tip of the iceberg of a vast floe of uncomputable in mathematics. Yet he fails to distinguish between 'unprovable (uncomputable)' and 'unknowable'. Godel's/Turing's Incompleteness theorems provide a glimmer that they are not one and the same. What is our program for studying the difference between the two?
Here's a conundrum: Chaitin asserts that 'omega', the fraction of Turing machines that stop, is 'maximally uncomputable', and that the derivation of n bits of omega requires n bits worth of axioms as input. He apparently deduces that this must, of necessity, be a fruitless pursuit, with an unvoiced argument that must go like this: "since axioms are 'arbitrary' and 'random', if we need to create N of these arbitrary, random statements to get N bits of a number, then we are in effect creating random numbers out of thin air." But yet, in real life, there are many axioms that humans are familiar with and 'know' that are not 'random', but are meaningful. By leveraging these axioms, and then a few more, could we not potentially expose a few bits of omega, asserting them in some 'obviously true' fashion, without being able to prove any of them?
Other interesting corners of the noetic universe lie near the tension of mind-body dualism, in particular as it pertains to ordinary Christian beliefs. Ordinary Christians are called upon to believe, and to take as true, the divine nature of Christ. This is, of course, very different than the fact that scientists and engineers believe and take as true various 'scientific' facts. Many scientists are endowed with the intellectual capabilities of observing and verifying the objective reality of scientific facts. Religious faith is the request to make beliefs into truths, unanchored in objective observation, but rather in subjective, experiential mysticism. It is anchored in the mundane 'gut-feel', in the mild trances and epiphanies of religious service, and possibly also more deeply in the Zen enlightenment or the Christian mystic experience. (Although it is easily argued that the Christian mystic experience does little to support the objective belief in Christ: the mystic cannot share the experience with the pedestrian believer, and rather, the lay believer is called upon to believe in the authenticity of the mystic experience.) The religious take faith and turn it into truth, they 'make it true' for themselves. In the above discussion, I use religion to provide the example because it seems to be one of the purest examples of the act of converting subjective experience/belief into a form of truth that is on par with humanist, objective truth (that is, to the faithful, the strength of belief, and the veracity of the claims is no less in the religious believer as in the scientist/engineer). By believing, truth is 'made to happen'. Truth comes into being through faith. It is an exercise of will to believe.
But why are we talking about 'will'? Because will lies at the very boundary between past and future: 'will' is what we use to effect changes in the objective universe in which we live. We cannot change the past, but through the judicious use of will, we can 'change' the future. But what does 'will' have to do with platonic reality? In will, we find the act of creation. In today's post-industrial world, we use will not only to move shovels of dirt from here to there, or to build highways and bridges, but we use will to create cyberspace. Of course, academics have been using will to create and shape the noosphere since before Plato and Aristotle, but it is today that more and more people are aware of the ability to shape the phantom universe of ideas (which 'cyberspace' is an example of), and the dramatic impact of this world on physical reality. (Of course, we've known for a while that words shape revolutions, that 'the pen is mightier than the sword', that the 'workers relation to the means of production' is what shapes history, and that when Atlas shrugs, the world stops turning. What is different is that we have, for the first time, e-commerce and chat rooms and web pages: a large and increasing part of the population involved in intellectual pursuits, toiling daily in the noosphere.)
But now that we are here, isn't it time to get a better understanding of the physics of the noosphere? We understand the physics of stars and atoms and transistors. We do not understand the physics that makes the past unalterable, and the future unknowable. We understand the physics of light streaming through a window, yet we do not understand what it means to 'live in the present'. We use time in mathematical calculations, yet we do not know why time flows. We do not know how time flows. We know, but do not understand how, the exercise of free will changes outcomes. I believe that in this crux of past/future and free-will also lies the doorway to platonic mysticism. The uncertainty of the future, and free-will are intimately tied together. In turn, free-will and subjective experience and knowledge are tied together. Somehow, the flow and ebb of time is tied to the workings of the mind. And the mind ultimately works inside the domain of platonic reality. As Penrose points out, we just 'know', we can just 'see' that Peano's axioms 'make sense'. This is a form of subjective knowledge, and we must ask how it is that this form subjective knowledge is distinguished from other forms of mystic knowledge. We must also acknowledge that any and all subjective knowledge, rational or insane, has active repercussions in the physical, objective universe. That humans alter the world is not just a truism, but must someday be ultimately, rationally explained in the language of physics, in terms of atoms and electrons.
When Reuben Hirsch apologizes that mathematics, economics, religion and law are social creations, he is ducking the point: the world of traffic tickets, weddings, and TV news spots is not just social, but also physical. The social world is a particular unfolding of history and human interaction: of course, my life, and yours, is in a certain way determined by who is President of the United States. But my life could have been different if Columbus hadn't discovered the Americas. Of course, Mathematics and its social discourse, would have been different if Euler and Lagrange had never lived. Of course, Mathematics, like Art, has a history. Of course, mathematics, as taught in schools and universities, is inextricably connected to the history of Human mathematical endevours and the social contexts of these endevours. But this in no way disproves the existence of a platonic reality.
In a deeper sense, the history of politics and social interaction and mathematical research are mere accidents along the road of reality: they are 'merely' the realization of a particular set of events, which 'may have come out otherwise'. To a Buddhist, these events are irrelevant: circumstance is just part of what must be wholly and entirely rejected. By rejecting the specifics of any given life history and social context, one passes to Zen enlightenment. But it seems to me that the platonic realities sit in this same place as Zen enlightenment: detangled, in a peculiar way, from the dirty facts of life. My writing here is a deeply social act: I hope that others will read it. My real social life is miserable: rather than going out and partying, I set here in front of a computer, key-whapping. Life is what happens while you are planning other things. But this way of thinking only detracts from the true memetic nature of this text: the thoughts contained herein address, correct, and amplify other memes. This particular meme might not exist if I'd done something else this evening; this in no way denies the existence of memes. But it is obviously clear that memes need a social fabric to breed and exist, much as minds need brains. To cavalierly dismiss mathematics as a social invention is to deny the existence of memes. Its akin to stating that thinking is just brain activity: that it is indeed, but its also something more.
As Don Cupitt (*) and other post-modern thinkers point out, it is the imprecision and fluidity of semantics laid upon the framework of the exactness of syntax that allows narrative to talk about and transmit ideas of the ineffable from one human to another. Indeed, it is language that lies between the subjective and the objective, converting noetic experience into words on a printed page, moving electrons in a wire, moving steam shovels. Clearly, computers have 'memory', and this is a consequence of the homomorphism between Turing Machines and Grammars (Lambda Calculus). Clearly, the concepts of computation are deeply and intricately involved in the function of human thinking. However, I claim (just as Penrose does) that the Turing machine in us just provides the solid, 'real', objective syntactic structure on which we can hang objective reality. Turing machines have neither free-will nor understanding nor self-awareness nor cognitive abilities; science has yet to deal with the semantic component of linguistics and it is in the semantic component that the ineffable and the mystic seem to lie.
The failure of the Reality Club's assertion that 'mathematics is an invention of the human brain, that mathematics is no more than a predisposition in our neural pathways to add and subtract numbers' is not so much that the statements about the brain are false, as it is rather due to the lack of appreciation for the physical basis underlying mind-body duality. In the modern, atheistic world, it may seem convincing to sweep away the belief of God into some neural predisposition, some curiosity of the biochemistry of the brain that makes the combination of wine, dance, meditation and chant lead to trances and epiphanies: just curious psychological phenomena to be studied. Yet it completely misses the point: studies of structure only reveal structure. Of course, if you look for a neural pathway for this or that, you will find it. What were you thinking, that such a pathway wouldn't be there? Finding the mechanisms or pathways of mathematical thought neither supports, nor denies the objective existence of platonic reality. The objective existence of subjective states may be viewed in PET tomographies of brains. But a PET scan neither proves or disproves the existence of the noosphere, any more than the discovery and description of a biochemical pathway does.
This is not to say that objectivism doesn't carry the seeds of its own destruction, for indeed, it does. Some day, maybe someday soon, some neurologist, possibly working in a vein similar to Stuart Hameroff, will uncover the link between the quantum world and the biochemical process of thinking. Some day, possibly during work with quantum computers, an inkling of the connection between quantum state reduction and free-will will be discovered. That day, we will have to reconsider fundamentally what it means for time to pass by, and we will have to reconsider what is meant by objective reality. The ability of the human mind to mold objective reality will no longer be a truism known to school-children, but will have empirical basis in scientific fact. Just because the subjective world is locked up in our brains doesn't mean it doesn't exist, no more than an unbreakable ciphertext (a mere string of random numbers?) doesn't mean that a plaintext doesn't exist.
P.S. The Reality Club is thoroughly and incredibly annoyingly elitist. To be a bit rude, but they can take that attitude and stick it where the sun don't shine. Wake up, this is the real world, already! Deflate that ego! Life is *not* a competition about who is richer, either intellectually or financially!