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I feel like the Kinks who reported in one of their songs that after their single "You Really Got Me" had become Number One people started ask them about their "politics and theories of religion" (Lola vs. Powermen album).
But since you asked, I found the Hersh interview fascinating. I will certainly look up Imre Lakatos on the net. I have some disjointed observations.
As to Math education, why is it so difficult to divorce the problem of teaching Math (or anything else) from the subject matter? To be sure, I feel I am in agreement with Hersh but I'd like to state this a bit more forcefully, because education is so important for society and because it is not a philosophical question. In fact it should be pure engineering, albeit social engineering. Try X, measure result, measure sensitivity to variations, vary in promising direction, repeat. We should be willing to teach even the "wrong" things (for example that there is no difference between numbers and numerals, or that numbers are necessarily base 10) if we get the desired result: happy customers, enjoying and skilled at useful math, many of them will explore deeper levels where the "wrong" things are easily "righted". My point is that the philosophy of Math, even a "true" one, is just a heuristic when used for pedagogical purposes, sozusagen outside its domain, and then not necessarily even a useful one. (I might argue that it is, in fact, pernicious).
It always fun to see old dichotomies being shown as empty puns, verbal crutches: internal/external, real/abstract. My feeling is that the current distinction between real (tangible) and abstract will slowly fade into a single concept (call it "necessity" for example) as we continue probe the depths of physics. At that point maybe we will identify in Math a "necessary" or Platonic part (also explaining why Math is so good at describing physics i.e. reality) and the social, Hershian part. I mean, the Platonic world would not be populated by arbitrary human constructs such as triangles, sines, Lie groups any more than physics needs "feet" as the measure of length or "red" as the measure of frequency any more - to be sure these were useful for a long time and are still useful in engineering. One-half and sine of pi over six are social games, but why not games rooted in some basic necessity just as our social existence is rooted in physical existence, especially if the physical and abstract existence will become to be known as the same thing?
I am contributing this in the spirit of brainstorming only. I do not necessarily believe a word of it.I apologise for not using the standard nomenclature-this is just a consequence of my ignorance.
- Charles Simonyi
I have to say that I disagree with almost everything that Reuben Hersh says. I can start with what I agree with, which is that the platonist and formalist schools of philosophy of mathematics do not capture what mathematics is. It must be said, however, that they are not stupid, or obviously wrong, I think that my disagreement with Platonism comes from two things: first from my philosophical commitment to the idea that the world we see is all there is, and that everything we see must be explained in terms of a network of relationships among real things. This leaves no place for a realm of real, eternal forms that transcends the particulars of the world, as well as no place for a view of the universe as if from outside of it. It also leaves no place for a platonic realm of mathematical form.
The second reason I disagree with platonism is that I think it is insufficient to make sense of the mathematical structures that arise in biology. It is one thing to speak of every possible platonic solid, but should we think that every possible biological species, or every possible niche, or every possible ecology exists eternally in some realm of ideal platonic biology? What about every possible way of earning a living in human society? Stuart Kauffman has been arguing that it may not be possible to list these kinds of things in advance, and I am tempted to think he may be right. There are lots of things that apparently cannot be classified in mathematics, like algebras or knots or four dimensional manifolds. For this reason, I suspect that Platonism will eventually come to be seen as insufficient to encompass the variety of possible mathematical structures. The point is that I believe that novelty is both possible and important, there are novel structures being discovered all the time, both by natural selection and human intelligence, and some of these are mathematical.
Formalism is easier to put away; it was basically killed by Godel's theorem. So what then is mathematics? I believe I understand the reasons why Hersh makes the move he does: that it is a shared construction of human beings, for that is some of it. There are an infinite number of possible mathematical structures, why some have been intensively thought about, while others were either thought uninteresting and most have not even been conceived of is a historical question. So historical and social questions may plausibly play some role in understanding why mathematics is as it is now. But this is not the same thing as to ask, what is a number, or what is mathematics.
I do not have an answer to this question that satisfies me, although I have thought a lot about it. In my opinion it is one of the really hard questions, like consciousness, or whether time might have begun, or might end. There are questions that I believe we cannot even conceive of satisfactory answers to given what we know presently. This does not mean they may not someday be solved-I think they may. There is a list of questions we are on the verge of solving, like the origin of life or the nature of space, that require twentieth century physics and mathematics, and that a nineteenth century person could not even have gotten started on.
Having said this, there are two thoughts that I find interesting when I try to think about what mathematics is. The first is the observation that time may play an essential role because a mathematical paradox can become a feedback loop when time is introduced. Something cannot be both true and not true eternally, but it can be alternatively in time. The second is the possibility that category theory may have profound implications for the question of what mathematics is, because it puts the emphasis exactly on relationships between different things. One might have looked down on category theory some years ago, but given the profound insights it has introduced into the relationships between different mathematical structures such as algebra and topology it seems very worth thinking about.
- Lee Smolin
From: Stanislas Dehaene
THE AMERICAN PI VERSUS THE FRENCH PI?
What is a number? As a neuropsychologist studying how the human brain wires itself to do mathematics, I'd answer that number is a parameter of our physical environment which is extracted and processed by dedicated cerebral networks-just like color, which is a subjective property entirely made up by brain area V4. Indeed, in my forthcoming book "The Number Sense", to appear in September, I show how animals and infants have a largely innate intuition about numerical quantities and their properties. Recent experimental evidence suggest:
I thus agree with Reuben Hersh that Platonism, or the view that mathematical facts are abstract and independent of human existence and knowledge, is not a tenable position. (My own neurobiological interpretation is that Platonism is a cognitive illusion that imposes itself upon so many great mathematicians because with training, their brains develop a vivid, seemingly real, internal image of mathematical objects. Presumably, one can only become a mathematical genius if one has an outstanding capacity for forming vivid mental representations of abstract mathematical concepts - mental images that soon turn into an illusion, eclipsing the human origins of mathematical objects and endowing them with the semblance of an independent existence.)
Mathematics is indeed a product of the human mind and brain, and as such it is indeed a very human enterprise, fallible, revisable, and highly dependent on the limits and abilities of our cerebral equipment. Does that mean, however, that mathematics is a purely social activity? The trouble with labelling mathematics as "social" or "humanistic", and with comparing it to art and religion, is that this view completely fails to capture what is so special about mathematics-first, its universality, and second, its effectiveness. If the Pope is invited to give a lecture in Tokyo and attempts to convert the locals to the Christian concept of God as Trinity, I doubt that he'll convince the audience-Trinity just can't be "proven" from first principles. But as a mathematician you can go to any place in the world and, given enough time, you can convince anyone that 3 is a prime number, or that the 3rd decimal of Pi is a 1, or that Fermat's last theorem is true. The point is, universal agreement is often easily reached about what constitutes a mathematical fact. This makes a unqualified relativistic, social, Lakatosian, or post-modernist view of mathematics totally untenable (it is unclear to me whether Reuben Hersh himself adheres to such an extreme relativistic point of view). The ridicule of this position has been recently pointed out by Sokal in "transgressing the limits...". Relativists notwithstanding, the value of Pi does not vary from culture to culture, nor does each culture have its own different "mathematical universality", as I recently heard in a post-modernist talk last week in Paris!
The other key difference between math and other cultural objects such as religions is its effectiveness. This was, and still is, a subject of awe and wonder for physicists like Wigner and Einstein. "How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?" Einstein asked in 1921. This is bound to remain forever a mystery as long as you adhere to a strong relativistic position, which asserts that mathematics is the result of the arbitrary cultural choices of mathematical "churches". For that matter, indeed, the effectiveness of mathematics is also not easy to explain if you believe, as Hersh seems, that mathematicians pursue their work for the sole purpose of its abstract beauty.
My tentative solution to both of these riddles appeals to evolution-of the brain and of mathematics. In my opinion, mathematical objects are universal and effective, first, because our biological brains have evolved to progressive internalize universal regularities of the external world (such as the fact that one object plus another object usually makes two objects), and second, because our cultural mathematical constructions have also evolved to fit the physical world. If mathematicians throughout the world converge on the same set of mathematical truths, it is because they all have a similar cerebral organization that (1) lets them categorize the world into similar objects (numbers, sets, functions, projections, etc.), and (2) forces to find over and over again the same solutions to the same problems (I am reminded of the reinvention of place-value number notation in 4 different cultures widely separated in space and time-chinese, babylonian, maya and indian). The common structure of our brains can explain why there is trans-cultural convergence in mathematics despite different social perspectives. In that respect, mathematical objects, while they are human constructions, are radically different from other cultural constructions of Western societies such as Christianism, Nouveau roman, symphony orchestras or French cuisine. Pies come in all sorts, American, French or Vietnamese; but hopefully there'll always be only one number pi!
- Stanislas Dehaene
From: W. Daniel Hillis
I certainly cannot argue with Hersh's premise mathematics is a part of human culture and human history, but surely it is also something more. Mathematics is not just a game or a poem with its own set of internal rules: it also has a striking correspondence to the real world. If I follow the rules it tells me things about the real world. A calculation can tell me where a ball will go next, what shape the bubble will be, or when the train will arrive. Often a mathematical construct is invented long before the corresponding reality is even noticed. Dirac, for instance, suggested the existence of anti-matter just because the equations of quantum mechanics also allowed for a negative solution. As far as I know, this magical connection between the abstract operations of mathematics and the real world remains entirely unexplained, but is surely an important part of what make mathematics special.
From: Jaron Lanier
There is room for a pseudo-humanistic philosophy of mathematics that seems more true to me than either Hersh's approach or strict Platonism as he presents it. In this philosophy, the particulars of math would be understood as platonically mandatory and eternal, but the range of possible areas of math to study and know would be understood to be breathtakingly large. So large that two different cultures undertaking mathematical study might not necessarily come across any common material. It is hard for us to imagine aliens not thinking about integers, but it is not logically impossible. There are some elements of logic itself that would have to crop up in some form, but the notion of what form would be most elegant and normal could be so variable as to leave room for a universe of virtually disjoint cultures of mathematics. Cultural diversion in math is more pleasant than in other areas, since it will never lead to authentic contradiction. This approach gets out from under the common, but false, implication of determinism in the history of mathematical inquiry that weighs down the teaching of a subject that should be joyous like music. It allows educators to treat math as culture, but at the same time avoids relativising the one area of human activity in which we can know truth.
From: Reuben Hersh
To: Charles Simonyi, Lee Smolin, Stanislas Dehaene, and W. Daniel Hillis
It's interesting that even though Platonism is the most popular philosophy among mathematicians, all four respondents readily reject it.
I have not solved all the philosophical dilemmas of mathematics. Especially, Wigner's dilemma of "unreasonable effectiveness." I have tried simply to give an honest account of mathematical activity in real life, without any dogmatic preconception. That includes recognizing the science-like aspect, the reproducibility and near unanimity of mathematical results, including the value of pi.
How does a socially shared concept system possess these science-like qualities? Kant asked, "How is mathematics possible?"
His ingenious answer doesn't work. There are no universal intuitions of time and space. But contemporary work in neurophysiology, like that described by Dehaene, may in a sense revive Kant's idea, now based on empirical science rather than pure speculation. If we do have brain structures for counting and for certain elementary spatial properties, we have them because they have survival value. They correspond to physical reality.
Now, certain full grown mathematical theories that physicists use also can be said to have survival value. Not biological survival, but social survival. (It has been argued that Newton's celestial mechanics gave England an edge in marine navigation, which would have been an edge in naval and commercial rivalry.) Can this parallel explain why mathematics is what it is?
If you look into the problem in more detail, this explanation is not so convincing. For instance, Heisenberg found matrix theory ready to hand for his version of quantum mechanics. Matrix theory was originated long before by Cayley, who found it a nice way to think about his algebraic transformations. Do you believe the military or commercial survival of England was really behind Cayley's thinking?
Complex numbers come in handy for a lot of things, like alternating current calculations and a scalar field for Hilbert space in q.m. Survival value, yes, if a.c. current and quantum mechanics do increase somebody's survival chances. But only after the fact! Originally, they were just unwanted "false solutions" for certain quadratic equations. Later, for cubics, they came in, uninvited, in the formula for real solutions. Were Ferrari and Cardano unconsciously getting ready for Steinmetz's electrical calculations?
This reminds me of anthropic discussions in cosmology. How in Heaven's name could it happen that the values of the fundamental constants are just what they need to be to make human life possible?
How is it that by solving problems, and inventing tools and concepts to solve those problems, and then solving the new problems about those new tools and conceptsŪmathematicians often give physics a hand?
Naturally it's no surprise that mathematicians working on questions from physics may give physics a hand. But that's not where fiber bundles and connections came from.
It's a mystery. I haven't tried to solve it. Is it more fruitful to be hung up on this mystery, or to accept it and go ahead?
You can't explain why there is matter rather than nothing. You don't wait to answer that before you do a little physics.
I can't explain (in a detailed, rather than vague and general way) why the social activity called mathematics is possible. I can recognize that since it exists, it is possible, without Platonistic ghosts or formalist devaluation. I can try to watch it with understanding, try to see what it does and how it works.
I agree largely with Dehaene, in particular with his psychological explanation of the origin of Platonism. I think the difference between his view and mine is largely a matter of emphasis.
I do not believe and would never say that mathematics is merely the pursuit of abstract beauty, or a game not rooted in the human mind and the physical world. But first of all, it's a problem-solving, theory-building activity, carried out on the basis of a given theory, and elaborated in the judgment of peers. Rigorous logic and physical application are usually far in the dim distance. To try and understand this activity, this world, out of its social context, is to make it incomprehensible, or comprehensible only by a falsification.
There are three serious objections to accepting the status of mathematics as part of the socio-cultural-historic level of reality.
Point 1 was taken up in my original interview. But I will repeat, briefly. "Nine" or any other number word has two usages, as adjective (describing a physical or other collection) and as noun, describing something independent of any particular realization, a general concept, usually referred to as part of an abstract structure, the natural numbers as governed by certain rules (axioms). The number-adjectives have no least upper bound, yet it is easy to write down a numeral for a number greater than any of themŪthat is, that will ever be counted or observed. In the shared concept of abstract theory if you will, that is a contradiction
(To Charles Simonyi:) There is a 2-volume collected papers of Lakatos, but his masterpiece is a separate book, "Proofs and Refutations."
Comments and arguments are invited.
As always, very smart guests, fascinating subjects. I was especially fascinated by Stanislas Dehaene's comments on the existence of dedicated neuronal circuits in the brain for processing numbers, since I believe that present computational theories may have to be extended, in the way that Einstein extended Newtonian mechanics, so that computers may begin to approach the richness of ways of *computing* that the human brain possesses today.
First, a mild disagreement regarding the sentence "That this 'number sense' is also present in animals, and hence that it is independent of language and has a long evolutionary past." Starting out at the level of neuronal polymorphism, and looking at the interesting clinical examples of such, (case Alex for example in language acquisition), perhaps the division drawn between mathematics and language is one of hierarchy rather than a "logical" one. At what point of ancestral connectedness would you say that there is "independence? Naturally, this may all be in your book, which I eagerly await.
Have you found a connection between linguistic ability and mathematical one that is closer to that of either one to intelligence itself? But back to the dedicated circuits in the brain to process numbers, to what extent does this eminent group gathered here by John Brockman see these discoveries affecting the computing industry? Personally, I am hopeful that engineers who followed the experiments of the so-called "chemical computers" (Oliver Steinbock, Agota Toth and Kenneth Showalter at West Virginia University...) may see, as did Showalter (see Peter Coveney and Roger Highfield in their book Frontiers of Complexity), the implications of path optimization in neuron networks.
My interest is in the path optimization implications as related to a machine that I will call the "entropy pump machine". An entropy pump machine acts as a semi-permeable membrane between regions of different relative entropy, always maximizing the gradient between the two. This is obviously a path optimization phenomena. I hope that the next generation of computers will rely on some chemical at first, then neuronal architecture, for then not only will we have vastly superior computers, but we will also be solving many evolutionary and philosophical ones too.
From: George Johnson
First I'd like to say hello to Reuben Hersh, my New Mexico neighbor, whom I've read with great delight but not yet met.
This question about the ontological status of mathematics can be applied to physical laws. Do they exist only in the neural encodings of the human species or are they universal, hovering in some platonic phantom zone? I wonder if information-physics can be used to forge a comfortable middle ground. In Fire in the Mind: Science, Faith, and the Search for Order (http://www.santafe.edu/~johnson/fire.html) I put it like this: In Zen and the Art of Motorcycle Maintenance, Phaedrus, the author Robert Pirsig's alter ego, is sitting outside a motel room in the West, drinking whiskey with his traveling companions and listening to his son, Chris, tell ghost stories. "Do you believe in ghosts?" Chris asks his father. "No," Phaedrus says. "They contain no matter and have no energy and therefore, according to the laws of science, do not exist except in people's minds. Then he pauses and reflects: "Of course, the laws of science contain no matter and have no energy either and therefore do not exist except in people's minds."
Pushed up against this edge, science often retreats into platonism. Here on earth there may be no such thing as a perfect circle, but we recognize the rough approximations because we somehow have access to the perfect Circle, a pure idea existing in a separate ectoplasmic realm. And so we are left with a duality between mind and matter, ideas and things.
Some followers of the information physics being pursued in Los Alamos, Santa Fe, and elsewhere suggest a way of bridging the divide: the laws of the universe are not ethereal, they say, but physical -- made from this stuff called information, the 1s and 0s of binary code. And so they seek to turn science back on itself and use information theory to understand where the laws of physics lie (in both senses of the word -- where they reside and what their limits are). If information is as physical as matter and energy, and if ideas and mathematics are made of information, then perhaps they are rooted in the material world. But the price for banishing platonic mysticism may be a dizzying self-referential swirl: the laws of physics are made of information; information behaves according to the laws of physics. Everything begins to seem like ghosts.
From: Ian Stewart
One point that did occur to me concerns the argument that because there were nine planets around before humans came on the scene, that means that the number 'nine' in effect existed already.
I don't agree. The problem is that the universe has no concept of 'planet', and in a sense would not know what to count. It is humans who classify the billions of objects floating around in the solar system according to size and shape, and decide that some are planets and others not. Without this classificatory step, there is nothing to count. In short, until something selects a set of things, there is no preexisting concept of number.
This is not to say that I agree with Reuben that maths is purely some kind of human activity. (I'm not sure that's what he meant anyway.) I imagine, for instance, that other intellligent beings -- if they exist, which I think likely -- could develop a similar kind of system. Howewver, I dopn't see why it has to be similar to ours, except in 'style'. For instance, aliens that take the physical form of plasma vortices in the atmosphere of a star might know far more magnetohydrodynamics than we do, but have nbo concept of 'triangle'. I think their maths would be in the same philosophical area as ours, but might not mesh terribly consistently.
I was told that when Jung invented the idea of the collective subconscious, a quirk of the German language left it ambiguous whether he meant a single pool of knowledge into which all humans dip, or a collections of separate -- but very similar -- pools. When we use a single word 'mathematics' we seem to be thinking of a single collective pool -- but, like all human knowledge and thought, it is really a collection of separate pools. Because maths is very precisely determined, the education system can ensure that all those pools overlap in a very consistent manner, giving the illusion of a shared common pool. I think this is the sense in which maths is basically contained in human minds. BUT there is a more abstract sense in which it is the common pool -- leading a virtual existence but feeling just as real to each of us as we dip into our part of it as a physical experience does -- and that's a more interesting kind of existence. It's why many mathematicians think platonistically. Medicine, say -- and definitely music -- is not as tightly constrained to be consistent, so is NOT really the same kind of thing.
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