A Mathematicians Apology: 0 (Canto Classics)

Canto Classics: A Mathematician's Apology

Recent exceptions to this include Brown ; D. Pedersen ; Grialou, Longo, and Okada ; Norman Wilson and Grosholz emphasize the conceptual drift and productive ambiguities involved in our applications of mathematical diagrams and representations. To be sure, Wittgenstein forwards many criticisms of mentalistic appeals to self-evidence or intuition or interior states of direct insight, and he rejects the idea that configurations of signs or diagrams can interpret or apply themselves, referring to an abstract reality on their own.

In his writings on mathematics he articulates these criticisms partly by appealing to the well-known, broadly constructivist thought that the algorithmic, explicitly rule-governed, humanly calculable and controllable elements of mathematics e. However, for Wittgenstein even in the Tractatus mathematics involves more than simply rules or calculations: To know a result in mathematics is to be able to see it portray it, communicate it as necessary, inevitable, unsurprising, in the context of proof or calculation, or definition, or method of argumentation, or representation, or discussion of the quotation from Augustine in PI and his remarks about Hardy Wittgenstein to Schlick July 31, , quoted and discussed below.

On appli- cations of mathematics and aspect perception in the pre-Tractatus period, see Floyd II, ; NB He is not denying that certain interesting matters may be hidden, but recasting our ideas about what kinds of hid- denness there might be, and what kinds of openness and revelation mathematics, logic, and philosophy might, on a realistic view,10 provide. Anything else might seem to risk falling into psychologistic mentalism 8. Two relevant discussions of tensions facing debates about the ampliative or nonampliative character of deduction are Dummett aand Dreben and Floyd It is true that Wittgenstein is often reflexively suspicious of his own appeals to aspect talk on the significance of this general point, see Baz However, he keeps returning to talk of aspects throughout his life, as if he can never quite rid himself of the focus such talk requires.

Instead, I propose that we also regard Wright , , emphasizes that phenomenology has little hope of answering the con- cerns inspired by, e. This may explain why interpreters have repeatedly tried to provide such a basis inspired by Wittgenstein. Wright , , Marion , are correct, I believe, in taking Wittgenstein to be focusing on a loose, indeterminate form of finitism as a style of mathematical practice, rather than strict finitism of the principled kind broached by Bernays, Wang, and Kreisel and discussed in Dummett b. How precisely RFM I is indebted to Frege is an interesting question, which it would be use- ful to explore.

The idea is partly programmatic, and was applied directly to, for example, remarks of the mathematicians K. Our interests are revealed in the ways we express ourselves with con- cepts, and Wittgenstein is interested in what interests us cf. I am not going to argue that Wittgenstein forwarded a worked-out, systematic, philosophy of mathematics, for I believe he did not.

What he did do was to point, suggestively, toward a range of specific ways in which talk of mathematical experi- ence and practice, even talk of intuition, might make sense. At least he was right to raise a series of questions about the logicist idea that Frege and Russell refuted, as a priori irrelevant to philosophy of arithmetic, all possible talk of mathematical [ ] Wittgenstein and the Philosophy of Mind His remarks on mathematics turn on folding features of our descriptions and vocalizations of our experiences into the description of what mathematical practice is, on grammaticalizing the intuitive.

The process of calculation brings about this intuition. By investigating particular examples of proofs and diagrams and notations in logic and mathematics Wittgenstein is, at bottom, proposing a new kind of criticism directed at our paths of interest, appreciation, and preoccupation when we discuss mathematics in phi- losophy.

He is proceeding on the assumption that there are qualitative ways of exploring our talk, experiences, and activities within logic and mathematics as part of the ordinary, the everyday, the familiar, and, hence, as talk that may be repudi- ated, miscast, misapplied, and misunderstood when we philosophize. Surprise is neither simply an experience, nor simply an attitude or point of view on the world—although it involves and reflects elements of each of these. Adam Smith was right to distinguish between wonder, surprise, and admiration in his history of philosophy.

Although he allied himself, like Smith, with the ancient Greek idea that philosophy originates in wonder, rather than in surprise cf. Wonder, Surprise and Admiration, are words which, though often confounded, denote, in our language, sentiments that are indeed allied, but that are in some respects different also, and distinct from one another. What is new and singular, excites that sentiment which, in strict propriety, is called Wonder; what is unexpected, Surprise; and what is great or beautiful, Admiration.

We are surprised at those things which we have seen often, but which we least of all expected to meet with in the place where we find them; we are surprised at the sudden appearance of a friend, whom we have seen a thousand times, but whom we did not ima- gine we were to see then.

We admire the beauty of a plain or the greatness of a mountain, though we have seen both often before, and though nothing appears to us in either, but what we had expected with certainty to see. Surprise, therefore, is not to be regarded as an original emotion of a species dis- tinct from all others. The violent and sudden change produced upon the mind, when an emotion of any kind is brought suddenly upon it, constitutes the whole nature of Surprise.

Its function is, however, not to be reduced to that of a description of a particular psychological state. A famous mathematician works on a result, cannot prove it, and someone else does. Now the readers are not themselves surprised at the simplicity, for they had no expectation in the first place, they had not grappled with a problem made difficult by an earlier point of view—unless they can work themselves back into the original problem context, and allow themselves to be struck by what was once new, but is now jejune or trivial or obvious.

There are many examples that could be collected from working correspondence among mathematicians, and I commend to the reader their collection and examination.

They are also pos- sible experiences, and matters of normative judgment. In this sense the notion of surprise forms part of an account of the interest and significance of what we do, and why: He read the Proposition. By G—— sayd he he would now and then sweare an emphaticall Oath by way of emphasis this is impossible! So he reads the Demons- tration of it, which referred him back to such a Proposition; which proposition he read. That referred him back to another, which he also read. This made him in love with Geometry. This remark is repeated by the historian of mathematics e. The pleasures of surprise in the realm of the necessary are distinctive, and widely known.

Those who work mathematical puzzles in the daily newspaper are not focused on generating hitherto unknown truths, but on the distinctive pleasure of their morn- ing divertissement.

JF Das Ueberraschende: Witt on the Surprising in Mathematics | Juliet Floyd - www.newyorkethnicfood.com

Such capturing of our attention is in a certain sense ephemeral: But it is not right to say that if the generality of the sequence is to be mastered, every single step must come to be viewed as equally mundane and a matter of course. It is also that such mastery entails that we are able to draw con- trasts between the interesting and the jejune, the informative and the redundant, the noteworthy and the obvious.

Surprise is an initial rush18 of puzzlement or confusion that vanishes; Wonder asks, How is this possible? In its natural habitat, surprise inflicts a peculiar stamp upon human inten- tionality that is saturated with our values, preoccupations, concepts, prefer- ences, choices, and interests as we see them now. Like the boring, the trivial, and the beautiful, the surprising is an evaluative phenomenon. If this is right, then the phenomenon of surprise is not something that can underwrite any explanation of the mind-independent existence or ontological reality of eternal, unchanging, abstract entities, or any theory of the cognitively ampliative character of deduction, if only because nothing is, qua the thing that it is, intrin- sically surprising.

I say this because the phenomenon of the surprising in math- ematics—the very hardness of finding or understanding a proof of what one might not have expected—has sometimes been adduced as evidence of the mind-transcendent reality of mathematical objects. A fictionalist about mathematics can certainly point toward the fact that fictional narratives and characters may inform us of certain discoveries about the human, precisely by necessitating one or another way of looking at things, and not just things properly belonging to the fictional world. This may explain the attraction of psychologists and philosophers to surprise as something that may be investigated causally, in terms of bodily reactions; like laughter or the startle reflex, it has long been associated by philosophers including Wittgenstein with what is in at least some contexts merely psychological or bodily, quasi-intentional at best, brutely reactive, part of our animal nature cf.

The psychological literature I have in mind aims to portray the tapestry of human emotion as if it is woven from a finite palette of universal emotional substrata at least seven basic emotions, of which surprise is one , substrata mirrored in the muscular physiognomy of the human face. These substrata are seen as cognitive and evolved, largely automatic and uncon- scious, rather than conventional or culturally relative. For a discussion of the quasi-intentional, psychological state of the startle reflex connecting to the con- cept of surprise, see Robinson , and, more critically, Shanker , Shanker and Greenspan Radical step by step, stipulative conventionalism is another story.

As Diamond argues, however, Wittgenstein relies on the fact that we do not simply choose or stipulate what will compel or strike us: In these later remarks he has not wholly worked himself free from this line of thought, which he associated explicitly with the idea that the mathematician is not merely a discoverer [Entdecker], but also praise from Wittgenstein an inventor [Erfinder]. To this extent he is interested as much in how we talk about and what we do with surprise as he is in denying the concept any place in our discussions of mathematics cf.

My claim is that Wittgenstein is interested in what interests and occupies us, in our abilities and how we describe them, and thus his remarks on mathematics, like his remarks on aspect perception generally, are devoted to bring- ing discussion of our interests and values into a discussion of what our experiences of necessity are. That there can be misplaced articulations of our interests, needs, and abilities is clear enough; what I want to consider in what follows is the manner in which Wittgenstein aims to criticize such misarticulations.

With this sugges- tion I agree. Allow me first to discuss an example, drawn from finite combinatorics. Sudoku may be played with no knowledge of mathematics; these puzzles are solved by logic alone, although there are, interestingly, no words in them. Let us now consider the wider mathematical context that leads to an analysis of this game.

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There are only two Latin squares of order 2 see table 9. A suitable response lies beyond the scope of what I can write here. Perhaps Sudoku problems are logical in an extended sense of the term. Note that his analysis of tautologies in the Tractatus dispenses altogether with words or even particular variables and syntactic forms such as quantifiers, just as Sudoku problems do. That we do not presumably reflects the contingent but significant fact that most of us have been trained to quickly take in, distinguish between, and order the numeric signs with ease: Thus we are here viewing the numerals as at best indices, not as arguments cf.

These Sudoku signs contribute nothing to any proposition concerning cardinalities of sets or properties of the natural numbers. It belongs, so to speak, to a fictional world. Potter , , for the kind of underdescription that concerns me. The rule for generation is this: Our first example above is cyclic, as is the one shown in table 9.

In fact, as we shall see, once we step to a more general setting, we lose our focus on this concept. Another quick way to generate Latin squares of a given order is to take a permutation of the symbols 1,. As is common in mathematics, and as we have already emphasized, the signs Now we can ask, Are there essentially different i. To reflect on this question mathematically, we codify the notion of two Latin squares of the same order being as different as unlike or various in internal struc- ture as possible. A way to visualize this variety is to imagine superimposing one square on top of the other.

If we never see more than one array place at which each of the superimposed digits 1 through n match up, then the squares are as different as they can be. For example, for any pair chosen from the family of Latin squares shown in tables 9. The group of Latin squares in tables 9. In Euler, the most important mathematician of the eighteenth century, posed the following problem.

This question is easy to state: Well over a century after Euler considered the problem, G. With this precedent, for many years it was widely believed that there was, similarly, no pair of orthogonal Latin squares of order 10, the intractability of exhaustive enumeration for this higher order being regarded as a daunting obstacle.

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2. Canto Classics: A Mathematician's Apology : G. H. Hardy : .
3. Canto: A Mathematician's Apology by G. H. Hardy (1992, Paperback).

Now we can ask: Even today it is not known whether or not there are three mutually orthogonal Latin squares of order The daunting calculational difficulties in making progress on orthogonal Latin squares of these orders stands in stark contrast to great progress on other orders that depended directly on central mathematical developments in algebraic number theory in the nineteenth century.

Here is a beautiful application of field theory, allowing us to see Latin squares in a new way, and to see one system of mathematics in another. Z3, Arithmetic mod 3: These serve, I suppose, as a partial basis for generating many different Sudoku problems. This is an understatement. Since I first wrote this paper, the mathematics of Sudoku is now a burgeoning field of research. But, having encountered the process of generation of Sudoku through finite fields, one sees the situation with more refine- ment, as no longer governed by exhaustive enumeration of possibilities but by the way of thinking and calculating afforded by field theory see, e.

Once we bring in the algebraic interpretation of the problem in field theory, the dif- ficulties Euler encountered for the case of order 6 appear to be strikingly different from the simplicity of the case of the order of the prime 7 and of the orders of the powers of primes 8 and 9. This contrasts with the unknown terrain of order 10 Latin squares. We have come to see differences between the orders that matter to us, and that would have surprised Euler himself. If the structure of fields affords us a new way of looking at Sudoku, seeing Sudoku in the context of field theory affords us also a new way of seeing a wide variety of empirical problems—a Wittgensteinian point.

One can see this problem as affording an opportunity for the application of what we have just seen about Latin squares and orthogonality: Idiosyncrasies of these four subjects and order of dosage are assumed to be irrel- evant in this block design. Each particular combination of drugs will be adminis- tered twice, to two different subjects, thereby controlling for differing reactions of Mathematics must appear in mufti, i.

For it occurs where Wittgenstein is offering a revision or reinterpretation of the logicist demand that the semantics of number words make general sense of the applicability of number. This does not mean we have controlled for all possibly relevant factors, or that one could not design another experiment that would control for these same factors better, if need be perhaps over hundreds of trials.

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This shows that the suitability of any block design is subject to a restricted way of viewing the problem it is intended to solve. It also shows that our ordinary knowledge, our values, and our interests shape engineering and experimental problems. What is philosophically important is that we can and do shape the course of our experience here—the testing of the effects of the dosage itself—by choosing to arrange our experiences in terms of the theory of Latin squares. It is also a beautiful example of a surprisingly simple kind of solution by brute enumeration to a problem that appeared to be intractably complicated to deal with precisely because it was embedded in the context of higher algebra.

It is, finally, a beautiful example of a case In this passage Wittgenstein discusses a quite analogous phenomenon of application in con- nection with a diagram used to pose the problem of finding the number of ways one can trace every join in a wall continuously, and without repetition. This problem stimulated Euler to study what are now called Euler graphs, objects that are still studied today as part of the mathematical theory of graphs, a vast sub- ject that began with the seven-bridges problem.

Instead, what is of interest is that our way of looking at certain situations—both empirical and purely mathematical—is changed, and is changed for the more interesting. I want next to consider a passage from the great number theorist G. Hardy has been criticized for making this remark about the surprising, though not by Wittgenstein. So in this work he is aiming to give an account of what gives mathematics its ultimate life-and-death significance, not merely as a scientific subject, but as a pursuit to which he and others have devoted the better parts of their lives.

What is important is that he stresses, in this context, not the hard reality, but the beauty and aesthetic value of mathematics, its arresting character: We may take it for granted now that in substance, seriousness, significance, the advantage of the real mathematical theorem is over- whelming. It is almost equally obvious, to a trained intelligence, that it has a great advantage in beauty also; but this advantage is much harder to define or locate, since the Hardy is mentioned at MS , p.

Monk , ; compare King , 73, for testimony that Wittgenstein would meet with Hardy.

Wittgenstein, Nachlass , some apparently drawn from his annotations his own copy of the edition of this textbook. On the status and availability of these annotations, which I discuss in section 4 below, see n. I will not risk more than a few disjointed remarks. In both theorems and in the theorems, of course, I include the proofs there is a very high degree of unexpectedness, combined with inevitability and economy.

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