Abstract
Hersh (1997) in a book aptly named What Is Mathematics Really? stresses the great distance he detects between the reality of professional mathematical practice—contemporary and historical—and the reasoning in formal languages that philosophers (since Frege) have largely characterized mathematical proof in terms of. Hersh criticizes the reasoning-in-formal-languages view of mathematical practice and mathematical proof as “isolated,” “timeless,” “ahistorical,” and indeed, even “inhuman.” Hersh (1997, xi) contrasts this derivation-centered view of mathematics (and mathematical proof) with an alternative view that takes mathematics to be a human activity and a social phenomenon, one which historically evolves and is intelligible only in a social context. His alternative view pointedly roots mathematical practice in the actual proofs that mathematicians create—actual proofs that Hersh claims philosophers of mathematics often ignore.
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Notes
- 1.
Contrast, for example, discussions of which theorems have (and haven’t) been established in various branches of mathematics—standard and nonstandard—with similar discussions of what implications have (and haven’t) been established about one or another political policy (or moral theory). Another area where disagreements are notably largely absent (compared to other areas of discourse) is rule-governed games. It’s relatively straightforward to tell when the rules of a game have or haven’t been followed correctly. This, as I indicate in the course of this paper, is no accident.
- 2.
See Azzouni (2013a) for extensive discussion of natural languages (and by extension, of artificial languages) from a strictly nominalist perspective—that is, from a perspective that treats such languages as collective psychological projections.
- 3.
I was tempted by a position like this—in large part because of this analogy—until in 2005, when I realized that all the ways I was offering to bridge the gap between formal derivations in proprietary vocabulary (e.g., ZFC) and informal rigorous mathematical proofs weren’t likely to make formal derivations accessible to mathematicians in the ways needed for those derivations to explain the aspects of mathematical practice that the derivationist account needs to explain (see my discussion of this in Azzouni (2009a), especially section 16). I revisit this issue in section 9.
- 4.
By “open-ended,” I mean that the axioms from this family of formal systems, collectively speaking, aren’t recursively enumerable; if they were (and because the resulting systems are conservative extensions of one another), the axioms from these families of axiom systems could be coded into a single axiom system. Rather, they are found, when they are found, not by “a uniform process,” but by “essentially new methods.” Here I’m borrowing Turing’s language (Turing (1936, 139)). See the interesting discussion of this in Copeland and Shagrir (2013).
- 5.
See, e.g., Gilles (2013, 28) who presses these historical examples and others against my discussion of this in Azzouni (2006, chapter 6). Philip Kitcher has raised the same kinds of examples to me in conversation (November 21, 2002); and indeed, these illustrations often arise in the literature as support for the claim that mathematical proof and practices mutate over time.
- 6.
- 7.
Hilbert seems to have had a view much like this. See Posy (2013, 120-121), where he discusses Hilbert’s notion of “intuition.” In particular, Posy quotes Hilbert (1926):
[A]s a condition for the use of logical inference and the performance of logical operations, something must already be given to our faculty of representation, certain extralogical concrete objects that are intuitively present as immediate experience prior to all thought. If logical inference is to be reliable, it must be possible to survey these objects completely in all their parts, and the fact that they occur, that they differ from one another, and that they follow each other, or are concatenated, is immediately given intuitively, together with the object, as something that can neither be reduced to anything else nor requires reduction. This is the basic philosophical position that I consider requisite for mathematics, and in general for all scientific thinking, understanding and communication.
- 8.
I too have stressed this aspect of ordinary mathematical proof. See Azzouni (2005, 19), as well as other work of mine.
- 9.
It’s not insignificant that Frege and then Russell and Whitehead (and the early Wittgenstein) understood the formal languages they were studying as intrinsically interpreted—as meaningful. The practice of treating formal languages either as meaningless, or as entities to which a semantics can be attached (as it were) as an afterthought, comes much later.
- 10.
It’s, perhaps, not obvious that these latter two cases are as I’ve just described them—in particular that finite subregions of these fields have finitely many cells. (They may strike people, instead, as continuum-structured.) I make my case for this in section 5.
- 11.
Copeland (2015, 1) writes, “A method or procedure, M, for achieving some desired result is called ‘effective’ or ‘mechanical’ just in case: 1. M is set out in terms of a finite number of exact instructions (each instruction being expressed by means of a finite number of symbols); 2. M will, if carried out without error, produce the desired result in a finite number of steps; 3. M can (in practice or in principle) be carried out by a human being unaided by any machinery save paper and pencil; 4. M demands no insight or ingenuity on the part of the human being carrying it out.” (I’ve altered the typographical format of this quotation.)
I mean what I’ve described above to be in the spirit of what Copeland writes (which in turn is in the spirit of Turing (1936)), although as my discussion will indicate, I disagree with some of the details. Here are several points of clarification about these disagreements: “Symbol” is misleading, in the broader context of games—but also, perhaps, in relation to one of the intended applications, derivations in formal languages—so I’ve eliminated this word. Only in certain cases are the items being manipulated “symbols,” as I’ll discuss later; this is connected to the issue of “meaning” already raised in section 2. There is some redundancy in Copeland’s (1), (2), and (4); the idea is that something (without intelligence) can carry out the procedure. In any case, I’ve dropped (2) altogether (as well as the relevant phrase “for some desired result” in what Copeland takes to be defined). This is because there need not be a “desired result”; there is only the question of how to characterize the intuitive idea of a method or procedure being “effective” or “mechanical.” Goals needn’t come into it. Finally, I’ve generalized the cases of procedures to ones beyond those using pencil and paper, as Turing originally restricted the discussion. It changes nothing essential to do this (or so I’ll argue in this and the next section, following Gandy and Sieg); besides, since certain diagrammatic proofs now occur in the medium of computer graphics, instead of pencil and paper (or chalk and chalkboard), the generalization is needed for the topic of this paper: informal rigorous mathematical proof. Notice, in particular, that certain topological proofs can be animated visuals where, for example, one shape smoothly transforms into another over time. A last point: I’m unsure what role “exact” plays in Copeland’s characterization of the intuitive notion of mechanical or effective procedure. Many mechanical or effective procedures have (or can easily be imagined to have) rules that don’t dictate, given a game state, a transition. This may be because no rule applies to this game state or because it’s vague whether a rule applies or not. (If umpires can disagree on the outcome of a rule or whether it even applies, we stipulate that the rule doesn’t apply.) At this point, therefore, the game episode ends (“halts” is a perfectly good term for this). Perhaps what’s meant is that the admissible transitions of game state to game state that are determined by the rules should be clear to all participants.
- 12.
See Turing (1936, 135-136).
- 13.
- 14.
Sieg (2013, 190) writes: “The methodological difficulties [of there being no proof of Turing’s thesis] can be avoided by taking an alternative approach, namely, to characterize a Turing computor [a human executing a mechanical procedure] axiomatically as a discrete dynamical system and to show that any system satisfying the axioms is computationally reducible to a Turing machine. … No appeal to a thesis is needed; rather, that appeal has been replaced by the task of recognizing the correctness of axioms for an intended notion. This [is a] way of extracting from Turing’s analysis clear axiomatic conditions and then establishing a representation theorem ….”
I should add that although Sieg’s approach is illuminating; I have doubts that it changes the epistemological situation very much—although it does show that the Turing-Church thesis is really just a matter of applied mathematics. The possibility that the Turing-Church thesis is false translates (without residue) into the possibility of apparent examples of computation that fail to be appropriately characterized by the axiomatized notions we were hoping those applications would fall under.
- 15.
This can be relaxed to a recursive set of rules.
- 16.
I expect these limitations to remain in place for machines (and more generally, for robots) forever, although this is controversial in some circles.
- 17.
Gandy (1980, 125) explicitly excludes from his approach anything which is essentially an analogue machine.
- 18.
Postulate 1: To draw a straight line from any point to any point. Postulate 3: To describe a circle with any center and distance (Heath 1956, 154).
- 19.
Feferman (2012, 376) writes, preceding his discussion of a proof of the Cantor-Bernstein Theorem, “Let us now turn to infinite diagrams which can be visualized in full, in contrast to those of the preceding section, though they may also involve the iteration of certain constructions.”
- 20.
I must stress, however, that I’m not arguing that we can’t conceive of completed infinities—this is a claim that certain constructivists and intuitionists make, but not me. I am saying that however we do conceive of such things, we don’t do so it by directly visualizing them. I’m urging us, therefore, at least when discussing diagrammatic proofs, to take Descartes’ old distinction between imagining and understanding seriously. (See Descartes 1979, specifically his sixth meditation.)
- 21.
These diagrammatic proofs, because they refer to both infinite “diagrams” and certain more traditional mathematical objects, have been revealed to have an interesting intricate (semantic) structure that’s pretty much never made explicit in their exposition. If it were, there would have to be an entire sequence of theorems about “diagrams” preceding these proofs.
- 22.
I discuss this in some detail—especially with respect to how it enables mathematicians to abbreviate and shorten informal rigorous proofs—in Azzouni (2006, chapter 7), especially p. 149-150. One nice topic area where distinctions between use and mention are regularly ignored by mathematicians is linear algebra. (Proofs in this subject area would become insanely longer if they were rewritten to avoid use/mention errors, say, between the talk of operators on vector spaces and their properties, as opposed to the properties of the notation—matrices—that represent those operators.) One common motivation for use/mention errors in mathematics is that computations often rely on properties of the notation that need to be described in the course of a proof. So, mathematicians regularly (but informally) engage in semantic ascent and descent in the course of many ordinary informal rigorous mathematical proofs.
I should add that this kind of “slippage” in mathematical discourse between radically different sorts of “objects” that are nevertheless co-referred to by the same noun phrases is typical of natural language generally. Consider the use of noun phrases, such as “London” or “person” in sentences like the following: “London is so unhappy, ugly, and polluted that it should be destroyed and rebuilt 100 miles away” (from Chomsky (2000, 37)). Also see Pietroski (2005) on this matter.
- 23.
For us to discern nearby dots (pixels) as separate, apparently, they have to be at smaller angular distances from one another than our eye’s angular resolution. And the average of the latter, apparently, is around 1 minute of arc. That’s why we can discern more distinctions if we move our eyes closer to a computer screen or to a piece of paper a diagram appears on. (Of course, there are serious limits to this method of improving resolution—posed by, among other things, the nose). But a fun trick this allows is pixelating a computer-screen presentation of (well, nearly anything) by moving our eyes closer to the screen. (As we age, our ability to do this diminishes—which issad.)
- 24.
Contemporary microphysics is no longer friendly to this metaphysical picture.
- 25.
For further details about the relevant notion of convention in play here, see Azzouni (2014). I should also note that what we recognize to be conventional properties of elements in our visual experience (e.g., that certain shapes are words with meanings) can nevertheless be experienced involuntarily and automatically. See Azzouni (2013a) for further discussion of this.
- 26.
The actual geometry of the cells is empirically determinable (in particular how many cells each cell is adjacent to)—although that geometry is relative to certain factors. Given a fixed distance of the eye to the page, the geometry is determined by when distinctions can’t be made among points and lines and so on. This geometry differs from person to person and changes if the position of eye to page shifts. In practice, we always manipulate diagrams by shifting game pieces in multiple cells at, in effect, the same time (e.g., even when extending a line on the page by ever so little).
- 27.
See Azzouni (1994, 50-52) and the citations given there.
- 28.
For example, if the functions depicted include ones that are discontinuous (e.g., with rational values all at zero but with irrational values where the curve appears in the diagram), I should add that the nearly total conventional nature of the diagrams in diagrammatic proof is widely either overlooked or underestimated. In particular, it’s often assumed without argument (and without the realization that the claim is implausible) that facts about which mathematical objects the items in a diagram look like play a major role in some diagrammatic traditions (e.g., the Euclidean one) in determining what mathematical objects (abstracta) they refer to. See, for example, Giaquinto (2016), especially his discussion of Azzouni (2013b). Also see Giaquinto (2007), especially chapter 12, which, in part, focuses on a distinction between the differences in extent to which a diagram depends on resemblance vs. conventions of representation. The role of convention, however (here and elsewhere), is underdescribed because it’s so widely taken for granted that dots on paper, for example, visually resemble (mathematical) points and drawn lines visually resemble (mathematical) lines. They don’t, since the mathematical items are not visualizable at all for pretty much the same reasons that a Koch snowflake isn’t visualizable: something with no dimensions (or only one dimension) can’t be seen. How visual capacities are exploited by diagrammatic traditions in a way that makes diagrams so much easier to understand than language-based proofs is—at present—not well understood (although see Azzouni (2005) on inference packages for suggestions). In any case, talk of “resemblance” is a metaphor that doesn’t help. (Also see Avigad (2009) for discussion of the relevance of contemporary vision science to the analysis of diagrammatic proof procedures.)
- 29.
See Manders (1995, 2008) and Azzouni (2004). For a discussion of how fatuous concerns about the rigor of diagrammatic proofs arise from tacit shifts in the mathematical interpretation of the configurations of a diagrammatic practice or from a failure to recognize that the mathematical interpretation of a diagrammatic practice isn’t a matter of what the diagrams look like (but is conventionally stipulated) (see Azzouni (2013b)).
- 30.
“Mathematical intuition” was, and continues to be, a big deal in philosophy of mathematics. See, e.g., Posy (2013, 127) for discussion, specifically about Kant’s seminal (and influential) views of it.
- 31.
Pelc (2009) makes a big deal of this, with respect to finding computer-checkable derivations.
- 32.
I’m going along with this claim about the experience of ordinary informal rigorous mathematical proof for the sake of the objection; but the point needs serious nuancing before it can do the real work that’s needed here. That ordinary informal rigorous proofs routinely provide an “Oh, I see!” phenomenology is exaggerated. (See Feferman (2012) on this and Azzouni (2013c).) Perusing ordinary informal proofs is a strikingly heterogeneous experience and so is the corresponding cognitive phenomenology accompanying that experience. Some steps may be facilitated by algebraic maneuvering that one knows only by virtue of certain memorized rules and not by anything like a feel for an implication relation between the statements, or anything conceptual. Other steps (quite often) are simply taken on authority (“Oh, that’s probably right”). Even a whole, Oh, I see how this goes, may actually be fairly piecemeal in the real understanding it provides. It’s a perennial (and horribly damaging) philosophical myth that mathematical proofs involve that many conceptual or implicational connections that are even candidates for a priori connections. The same point holds of the phenomenology of logical inference itself. See Azzouni (2005) and Azzouni (2008) on this. I take the point up about the heterogeneity of mathematical proof, and develop its implications, in section 9.
- 33.
As I’ve mentioned, the first objection originally drove me from the “derivation-indicator view” of mathematical proof,” at least as a view that the role of the indicated derivations aren’t purely normative, but additionally are supposed to help explain Rigor, Correctness, Agreement, and so on. See Azzouni (2005, 2009a). The second objection is recent and due to Tanswell (2015).
- 34.
- 35.
- 36.
We could attempt an error theory here: mathematicians are laboring under the false illusion that their informal proofs can be “filled out” the same way. But this “save” faces tension because the formal derivation is supposed to be why they think the informal proof is valid. I should add that Tanswell (2015) presses the overdetermination objection against derivationists in a different way than I do here—he offers a dilemma that turns on whether the derivationist account is agent dependent or independent. Regardless, my response to the objection (however formulated) is the same.
- 37.
When coupled with one or another foundational program, a specific language-based vocabulary is required. This, in some cases, may yield derivations (unique up to size) that avoid Tanswell’s objection, but nevertheless will allow a stronger version of the too-long objection.
- 38.
See the discussion of “framework facts” with respect to the Euclidean diagrammatic tradition in Azzouni (2004, 125, and what follows).
- 39.
This is even true of a subject area as apparently restricted as number theory. Number theory, after all, isn’t just Peano arithmetic. Everything in mathematics comes into play—as the recent proof of Fermat’s last theorem illustrates rather dramatically.
- 40.
Although Rav (2007, 315), recall, summarizing his discussion of several examples of informal rigorous mathematical proofs, writes “I hold that mathematicians’ manner of reasoning and inferences are based on meanings and an informal notion of truth that a formal deduction calculus cannot capture” (italics his).
- 41.
Rav (1999, 16) notes that matrix theory isn’t axiomatized. That’s right—how could it be? Just about anything can occur within a matrix diagram (integrals, series of integrals of functions, etc.), and just about anything can be “done to” matrix diagrams or to sequences or series of matrix diagrams (powers of matrices, infinite sums of matrices, and so on). See, e.g., Gantmacher (1960, 1977) for details. (Also recall the discussion in section 5 of Feferman’s examples—the same kinds of phenomena are coming up in this subject area.) Notice how the heterogeneity of mathematical proofs follows from the case of matrices alone: all sorts of mechanical-recognizable proof procedures are imported into matrix theory by the mere importation of the notation for these things into matrix theory. (One can imagine, e.g., actual geometric curves, representing functions—or other diagrammatic objects—occurring in the boxes of a matrix.)
- 42.
Of course there are games that don’t involve algorithmic recognizability. “Game” is a notoriously broad word. I’m simply leaving those “games” out of consideration: the remaining class of games is still bewilderingly variable.
- 43.
Anything? Yes, the antecedent can, for example, yield a trivial set of consequences: everything or nothing.
- 44.
There is a bit packed into this and the last paragraph that I really can’t get into here. The relevant “talking points” are what I’ve called “the external discourse demand”—that required are shared notions of logic and truth across the mathematics and sciences that are brought to bear on one another (Azzouni (2010), 4.7–4.9)—and also, what might be called “the assertional model” of scientific discourse. That is, using scientific discourse to represent (even when idealizing) aspects of the world, and to draw consequences about one’s representations of the world, requires “detaching the antecedent.” One can’t do this by “conditionalizing” one’s results relative to assumptions (Azzouni (2009b), sections 1 and 2).
- 45.
One way to do this, however, is to introduce logical connectives corresponding to implication operators that are minimal with respect to the propositions they imply. See Azzouni (2005, 34–39) for discussion.
- 46.
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Azzouni, J. (2017). Does Reason Evolve? (Does the Reasoning in Mathematics Evolve?). In: Sriraman, B. (eds) Humanizing Mathematics and its Philosophy. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-61231-7_20
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