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On the Nature of Mathematical Concepts:
Why and How Do Mathematicians Jump to Conclusions?

A Talk with Verena Huber-Dyson

J.C. Herz, Reuben Hersh, Stanislas Dehaene, Margaret Wertheim, Brian Rotman, and Verena Huber-Dyson on On The Nature of Mathematical Concepts by Verena Huber-Dyson.

From: J.C. Herz

(Verena Huber-Dyson wrote in EDGE 34:) "That practice, familiarity, experience and experimentation are important prerequisites for successful mathematical activity goes without saying. But less obvious and just as important is a tendency to 'day dream,' an ability to immerse oneself in contemplation oblivious of all surroundings, the way a very small child will abandon himself to his blocks. Anecdotes bearing witness to the enhancement of creative concentration by total relaxation abound, ranging from Archimedes' inspiration in a bath tub to Alfred Tarski's tales of theorems proved in a dental chair."

This leads into a set of questions about immersion and suspension of disbelief, vis a vis media. And the questions are: 1) how active must you be in the construction of experience in order to reach that level of immersion. If you're painting, if you're probing a solution space, if you're exploring "the beautiful slack," daydreaming, you are the architect of that experience, based on very little in the way of outside stimulus-you construct the experience from scratch. With a book, it's slightly less so, but the same principle holds, because a series of squiggly shapes printed on a page is really a very abstract thing, and you have to construct the words, and from there, the concepts, the images, internally. Further along the continuum, you have something like instrumental music, which is a very richly textured sensory experience but still rather abstract in that its meaning isn't specified. And then you have music with lyrics, which is still uni-sensory. And then you have audiovisual media, starting with film, which subsumes your perception by dominating the context, and television, and finally the garden variety web page, which has no cerebral legroom at all.

I would argue that the Net in its text days was closer to a book-much further up the continuum of immersiveness than the current cruise-ship buffet of HTML offerings. And a videogame is somewhere between music and film.

And 2) If a higher level of construction (which is not to say focus, but rather a kind of zen mindfulness) is necessary to bring about that kind of immersion, then how many people are going to be either capable or willing to engage in it? It takes a certain amount of intelligence and, more importantly, a certain amount of trust-in yourself, in the situation-to put yourself in that state, to be receptive to that experience, especially on a frequent and/or regular basis. There are very few people who can do that, and most of them are under the age of ten, and even those are a minority-the daydreamers-and the ones who can hold onto it are fewer still.

Which may change, actually. I think that growing up with videogames, computers, etc. extends the limits of that group outward from the people who read voraciously, draw, etc. to the ones who can't or won't necessarily "make" things but are willing to explore other kinds of imaginary spaces because they're lured in by the eye candy and the hormonal rat buttons. Any way you slice it, the hours drift by, and they become comfortable with that kind of flow.

Which leads to a third question, about the daydreamers: Are they born or made?

From: Reuben Hersh

Huber-Dyson's posting is impressive in several ways.

I especially liked "The positive integers are mental constructs. They are tools shaped by the use they are intended for. And through that use they take on a patina of reality."

I found another remark provocative: "Conceptual visioning is an indispensable attendant to mathematical thinking."

The rational number line vs the real number line-how do you envision the difference? Does the rational line have a lot of little holes scattered everywhere? Isn't any vision bound to be wrong and misleading somehow?

How about some of the "monster" simple groups? If there is some geometry associated with them, isn't this understood only after the group has been understood, not in the process of understanding it?

I feel that Huber-Dyson's remark is correct, yet I am unable to pin down what it is really saying.

Perhaps "visualization" doesn't necessarily mean a visual picture, but just some concrete example or interpretation to which we know how to apply some intuitive thinking.

- Reuben Hersh

From: Stanislas Dehaene

I find myself in agreement with most of what Verena Huber-Dyson states about the mathematical mind. Non-symbolic processing is clearly crucial, and non-conscious (or subconscious) mental activity plays a considerable part. Jumping to conclusions, only later to go back and work out an exact proof, has been stressed by many mathematicians in the past, including Hadamard, Polya, Einstein and Poincare. For the most part, this conclusion is based solely on mathematicians' intuitions, but my research suggests that at least in the number domain, it can be validated by neuropsychological experiments. Yes, there is a non-symbolic representation of numerical quantities (which can be called "analogical" or "conceptual"). Yes, it plays a crucial role whenever we think about number MEANING, rather that merely do symbolic number crunching. And, yes, it can be activated unconsciously and give us "intuitions" about arithmetical relations among numbers as well as between numbers and space. It even has a specific cerebral substrate, so that in the near future we may hope to image its conscious or unconscious activation during (elementary) arithmetic.

I enjoyed Verena Huber-Dyson's dissection of the mental processes going on in her head as she was reflected over Ramanujan's finding (that 1729 is the smallest integer that can be expressed as the sum of two cubes in two different ways). Obviously, there are many different ways to "jump to this conclusion" (although a formal proof is relatively long). All of them seem elementary once they have been found! I still think, however, that my explanation is simplest because it only appeals to VISUAL recognition. Presumably, any mathematician knows that 1, 1000, 1728 and 729 are cubes (1728, I know realize, is obvious to any Englishman because it is the number of cubic inches in a cubic foot; 729 may not be so well known). Once you know this, it is VISUALLY obvious that 1729 is 1000+729 and 1728+1. No conceptual activity is required, although of course higher mathematics may be used afterwards to deepen understanding of this fact, as shown by Verena Huber-Dyson.

The whole point for which I used that particular example is that, at first, Ramanujan's feat looks super-human-but under my explanation, it is a feat that anyone with some interest in arithmetic facts can understand and could have performed! I thus fully agree with the following passage from Verena Huber-Dyson, which I think is worth emphasizing:

"Mathematics can be done without symbols by a particularly 'gifted' individual, like, e.g., Ramanujan. What that gift consists of is one of the questions raised in the EDGE piece. Obviously we are not all of us born with it. Nor do I believe that all people born as potential mathematicians become actual ones. Tenacity of motivation, an uncluttered and receptive mind, an unerring ability to concentrate the mind's focus on long intricate chains of reasoning and relational structures, the self discipline needed for snatching such a mind out of vicious circles, these are only a few characteristics that spring to mind. They can be cultivated. Experience will train the judgment to distinguish between blind alleys and sound trails and to divine hidden animal paths through the wilderness."

Stanislas Dehaene

From: Margaret Wertheim

As in other various other EDGE postings on the subject of mathematics V.H-B talks about math being an embodied phenomena. She noted that counting stems from physically embodied beings enumerating physical objects (such as pebbles). Lakoff has also stressed the importance of embodiment in areas like spatial perception, and Dehaene has noted that our brains seem to be physically wired for some sort of mathematical perception. All this suggests the beginning of a post-Platonist conception of mathematics, and potentially even of numbers. Perhaps, as Huber-Dyson hinted, even the integers are not the work of God (as Kronecker so famously remarked) but are intimately bound up with embodiment itself. I find it so provocative that many of the posters in the EDGE discussion on mathematics have been sidling up to the edge of Platonism, and looking beyond pure abstraction towards the idea of an embodied explanation of mathematics.

Apropos of this, I would like to note that mathematician/philosopher Brian Rotman has already formulated a powerful post-platonist and inherently embodied conception of numbers that meshes beautifully with the issues we have been discussing. Rotman has taken the implications of embodiment for mathematics very seriously indeed and has worked out a truly comprehensive post-platonist conception of "what numbers really are". His work goes far deeper than the Intuitionist or the Constructionist approaches and has considerable philosophical consequences for our thinking about mathematical objects.

This post-platonist conception of numbers is presented in his book Ad Infinitum: Taking God Out Of Mathematics And Putting The Body Back In.. The subtitle immediately signals the relevance to our discussions. In a nutshell, Rotman suggests that the integers do not have a separate Platonic existence, but only emerge from the (necessarily physically embodied) act of counting. In the absence of an embodied counting being, Rotman suggests that integers have no ontological validity. From this rather simple observation, he goes on to outline what amounts to a semiotic theory of mathematics. In his account, mathematics consists only of that set of objects and theorems that can be realized through a finite set of procedures, in a finite space and time, using a finite amount of energy, by a finite (i.e. physically embodied) being. Mathematics becomes, then, just like the other sciences, inseparable from the physical world. And NOT, as Platonists believe, a separate "transcendent" reality.

Quite apart from the major philosophical issue at stake here-Platonism having such a powerful psychological grip on western culture-Rotman's work has immediate consequences for our thinking about mathematical objects. In particular, if Rotman's vision of mathematics as INHERENTLY embodied is correct (as I believe it is, and as the work of Dehaene, Lakoff etc suggests), then mathematics CANNOT contain by definition any infinitistic objects-including (most importantly) the real numbers. Thus Rotman's philosophy of numbers provides an answer to the question raised by Reuben Hersh in response to Huber-Dyson. In his posting in EDGE 35, Hersh wrote as follows: "The rational number line vs the real number line--how do you envision the difference? Does the rational line have a lot of little holes scattered everywhere?"

Rotman specifically answers Yes to this question. Yes, he says, the real number line is not a continuum-it is, in effect, peppered with holes. One consequence of his post-platonist philosophy of number is that ALL infinitistic objects (including the irrationals) are idealizations that do not have any ontological reality. Since we cannot get to them by ANY realizable constructivist method in a finite amount of time, then they cannot be said to "exist". According to Rotman, such concepts as the irrationals (and as infinity itself) ought to be regarded as theological abstractions. As Rotman's subtitle suggests, his aim is to strip mathematics of illegitimate "theological" woolyness and ground it firmly in the physically embodied world.

As a corollary to the above, I note that the issue of embodiment is cropping up in a number of other EDGE discussions-Rodney Brooks, for example, insists (rightly I believe) on the necessity of physical embodiment for the realization of an "artificial intelligence". The COG project is founded on this premise. Embodiment, it seems, is a hot issue. It may therefore be of interests to EDGE readers to know that feminist philosophers have long been insisting on just this point-that knowledge and understanding of the world requires not just purely "mental" processes, but must also be grounded in the reality of bodily experience. As a feminist and a lover of science, I find it most interesting to see these two strands meeting up-though I wonder if many scientists are aware that they are now supporting a major claim of feminist philosophy?

From: Brian Rotman

I'd like to respond to some of the points made by Verena Huber-Dyson in her Edge article "On the Nature of Mathematical Concepts". I greatly enjoyed her elegant and richly imagined exploration of Ramanujan's encounter with cubes, but some of her more general remarks about the nature of mathematical activity require comment.

"Much mathematical reasoning is done subconsciously just as we obey traffic rules ... . Symbolic notation is an 'artificial aid' ... But it is not mathematics. Mathematics can be done without symbols by a particularly 'gifted' individual, like e.g., Ramanujan."

Of course mathematical reasoning (like every other kind) has subconscious aspects to it, but what is the passage from this to the claim that notation is artificial and not mathematics? And even if someone like Ramanujan could do mathematics without symbols (a proposition that strikes me as absurd on more than one level), what does this tell us about how any of us might do mathematics? Claiming symbols as artificial romanticizes mathematics as a mysterious and ineffable species of 'pure', i.e. linguistically untainted, thought; a claim that makes sense only if one is in thrall to a notion of language as a transparent and inert vehicle for the communication or transmission of 'thoughts' formed prior to or independently of it. If the debates in the humanities over the last thirty years have done anything at all (and to be on the 'edge' of thought but persist in a view of language untouched by them is surely odd) they've rendered such linguistic transparency untenable. In any event, the idea makes no sense for mathematics. The history of mathematics is impossible to tell except as an ongoing and highly complex interaction between writing (symbols, notations, diagrams, formalisms, ...) and thinking/imagining (ideas, concepts, intuitions, arguments, narratives, ...). For every occasion when mathematics appears as thought-driven, where intuition or conceptualization is seen as prior to its symbolization, one can find an example of the reverse effect in which new mathematics is created out of a diagrammatically or symbolically presented situation; for example, the discovery/invention of irrational numbers from the 1:1: 2 right triangle, the notation-driven formulation of -1 as a solution to an equation, the conceptual facilitations of category theory diagrams, and so on. In mathematics, language far from being neutral or inert is always inseparable from and frequently constitutive of the very objects, abstractions and relations it (subsequently) is seen to be 'describing'.

Having "... a tendency to 'day dream', an ability to immerse oneself in contemplation ..." as an important requisite of successful mathematical activity. Absolutely. I think Verena Huber-Dyson has hit the nail on the head with total accuracy, though anecdotes of inspiration in dental chairs and bathtubs, delightful as they are, seem to undercut what is, I'd urge, not merely a requisite but the very armature of mathematical thought. For some years now, I've been arguing exactly that by using a semiotic model, based on the writings of Charles Pierce, which understands mathematical reasoning/persuasion as a certain kind of waking dream or thought experiment.

According to this, mathematical assertions of fact are predictions about the mathematician's encounters with signs. A prediction is justified, i.e. a statement is proved, when (a suitably idealized version of) the mathematician propels a surrogate or proxy of his/herself around an imagined mathematical world, observes the result, and comes to the desired conclusion about what would have happened had the imagined journey been a real one. To get the model off the ground requires examining the role of imperatives (draw X, enumerate Y, consider Z, etc) mathematicians use to describe and communicate their work. Portraying mathematical reasoning as a particular kind of symbol-controlled Gedadenkexperiment proves to be very useful. Thus, at the time, I used it to explain how each of the three major philosophical characterizations of mathematics - formalism, intuitionism, platonism - was both undeniably attractive and fatally inadequate. Since then, I have developed the model to provide a new discussion of what it means to count and to examine what sort of metaphysical apparatus is folded into the ideogram '...' when we write 1,2,3,... to signify that the sequence of so-called natural numbers extends infinitely far.

That such a re-examination of '...' is not before time is evident from Ralph Nunez's comment, in the forum on Stanislas Dehaene's book, to the effect that the 'etc' symbol hides "a very complex cognitive universe" and that writing '...' after 1,2,3 is an "extraordinary cognitive achievement". Indeed it is, and one which until now has been masked by the idea, so famously expressed by Kronecker, that (wherever the rest of mathematics comes from) the integers come from God. A stance that is not only obfuscatory and lacking explanatory worth, but turns out to be directly challengeable. Moreover, challenging it has interesting consequences, since it leads to a notion of number, counting and arithmetic quite different from, and if anything more 'natural' in the age of the computer, than the picture of endless continuation familiar to us all.

Best wishes,

Brian Rotman

From: Verena Huber-Dyson
To: Brian Rotman

I am distressed by Rotman's accusation (April 11) of "romanticising mathematics as a mysterious and ineffable species....", which is, in fact, based on a misunderstanding. It was by no means my intention to call symbolisation "artificial". My mistake was to appeal tacitly to the rock climbing metaphor introduced earlier. The term 'artificial aid'-which I had deliberately enclosed in quotes-belongs to the technical language and refers to nuts and bolts, pitons and other hardware employed to extend the grasp, reach and security of the climber's bodily tools, toes, hands, knuckles... As far it was meant in my context I still consider the analogy appropriate and won't belabour it now. There is nothing mystical about the working mathematician's ability to perceive relations and perform operations on abstract structures in the mind, a mind that is as physical as limbs guided by a plan to scale a rock face. I had earlier referred to Brower's technical writings on logic, which are deep, compelling and clear without the use of the symbolism of mathematical Logic developed since then.

I still feel that laymen ought to be warned against a rash identification of mathematics with symbolic manipulation.

In the first place symbolism is meant to serve the purpose of communication, storage of insights and efficiency. That it cross fertilises concepts in the process of evolution is of course clear, accepted and appreciated. Incidentally the idea of mathematics, in particular mathematical logic, as the pursuit 'Gedankenexperimente' (glad to see that term accepted) was developed by the German logician Paul Lorenz in mid century.

If I ever get around to is I shall be interested to look at Rotman's books.

Greetings from Verena

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