Dialogue on Mathematical and Physical Knowledge


Physics is a meditation on the beauty of nature.  Mathematics is a meditation on the nature of beauty.


Here is a three-part dialogue on the origin of mathematical and physical knowledge.  Any resemblance between the fictional participants in this discussion and actual persons is purely coincidental.



Day One:  Clear and Certain Notions


Salviati: Our confidence in the various formal systems (PA, ZFC, etc.) and results in mathematics is necessarily based on an incomplete induction, just as is our confidence in any proposition of physics.


Simplicio: Nonsense! Our confidence in arithmetic is based on seeing a model: the integers N, addition, multiplication.  These are clear and certain notions, and the PA axioms hold in N; so we believe PA is consistent.  This is not a formal proof, nor can it be; but it is vivid and compelling.


Salv: You're describing the intuitionist point of view:  To "see a model", to apprehend clear and certain notions, and to simply know that those notions must be consistent.  For an intuitionist like Brouwer, mathematics has as its source an intuition that makes its concepts and inferences immediately clear to us.  You've gone further, implying that our chosen axiomatization is automatically entirely consistent with those intuitive notions, which the intuitionist surely doesn't suppose, but in any case, if you concede that this intuitive sense ultimately arises from (or is based on) multiple individual perceptions, all seeming to conform to a simple pattern, then you are subscribing to incomplete induction.


If, on the other hand, you maintain that our intuition is a single indivisible whole, arising not from any interactions with external objects but only from a pure metaphysical vision, then you are subscribing to Platonic mysticism.  In either case, exactly the same type of justifications can be (and have been) given for believing in various physical theories (cf Pythagoras, Kepler, Eddington, Einstein,...) 


The purpose of this discussion was to ascertain whether the way in which we "know" mathematics is qualitatively different from the way in which we "know" physics.  Nothing in the "I see a model" approach gives us any reason to think the two are fundamentally different. Both rest on necessarily incomplete induction.  Indeed it's been suggested that every formal system, if pressed far enough, is inconsistent.  Nothing guarantees us the existence of a consistent formal system with enough complexity to encompass arithmetic.


Simp:  What!?  You're not serious!  It's true that Arithmetic is quite complex, and in a certain Godelian sense we cannot hope to formally prove its consistency (within any formal framework whose consistency is more indubitable), but neither this nor anything else, I should think, establishes there are no consistent systems at all! 


Salv:  Calm down, Simplicio.  There is no claim to have "established" this result, but simply to point out that nothing guarantees us the existence of a consistent formal system with enough complexity to encompass arithmetic.  By the way, be assured that the existence of a contradiction in a formal system need not completely vitiate the system.  The only operative consistency in any system, mathematical or physical, is LOCAL.  We can define a metric on the space generated by the axioms of a system, and find that there is consistency within a certain region of that space, even though there are global inconsistencies.


Simp: In mathematics?  I don't think so.  Logics have been proposed that profess to 'tolerate' inconsistency, from what I hear; but logic as actually used in mathematics does not.  How would it go?... what metric?


Salv:  In any given fixed formal system the mathematics can be reduced to computation.  Within computation there are varying distances between pieces of information.  For example, given the integers x and y, the distance to the sum x+y is short, but given the integer N (=xy) the distance to the factors x,y is large (assuming P~=NP).  These distances can be quantified, as in polynomial time, exponential time, etc. 


Now suppose we have a set of axioms encompassing ordinary arithmetic, and just for fun I add one more axiom, stating that the 20 billionth decimal digit of p is 7.  What happens?  Does the system instantly dissolve before our eyes?  Can we now easily prove 1 = 0? Well, we might be able prove it, but it won't be easy, because (as far as I know) there is no short path between any existing theorem and the value of the 20 billionth decimal digit of p.  (Note that there are algorithms to produce the nth hexadecimal digit of p, and in some other bases, but none is known so far for decimal digits.)  I can assume it is 7, or I can assume it isn't 7, and it really has no demonstratable effect on any other theorem. This is an admittedly a clunky example, but it makes the point that it's perfectly possible to conceive of an axiom system that may well be inconsistent, but whose inconsistency is practically inaccessible.  (If 20 billion isn't safe enough for you, make it 10^10^10 or whatever.)  If I say that my axioms must be consistent because I've been working with them for many years and have yet to find an inconsistency, would this persuade anyone?


Is it conceivable that a more benign looking set of axioms might nevertheless imply two contradictory things about some computationally very remote entity?  Speaking of benign-looking axiom systems, we shouldn't overlook examples in mathematics, such as the naive set theory of Cantor, Frege, et al, that was developed extensively over a period of many years, with many meticulously proven theorems, before it was recognized that the system itself was ill-founded and inconsistent.


Simp:  Not Cantor.  Nobody has ever found something ill-founded or inconsistent in Cantor's work. 


Salv:  Your loyalty is admirable, but it's well known that Cantor's early ideas were ill-founded.  Even the humble "Encyclopedic Dictionary of Mathematics" (2nd Ed., MIT Press) tells us that


It was G. Cantor who introduced the concept of the set as an object of mathematical study...   Meanwhile it was pointed out that Cantor's naive set concept leads to various logical paradoxes...


Simp: Infallible, we surely are not.  Errors in published papers happen all the time; on occasion a mistake survives for years. However, I'm not aware of any established body of mathematics, however old, having been invalidated and overturned by later work, ever.


Salv:  Now, Simplicio, statements like these undermine your previous claim that incomplete induction plays no part in your confidence in mathematics.  Moreover, the statement is simply not historically accurate.  For example, Euclid's "Elements" was surely an established body of mathematics if ever there was one, displayed as a model of mathematical rigor and perfection for two thousand years, but today it's widely acknowledged to be shot full of gaps and errors. You may counter that nevertheless the sum of the squares of the legs of a right triangle equal the square of the hypotenuse, but that is just the raw data.  We could equally well say that Ptolemy's astronomy was never overturned, because after all the sun still sets in the west.


If you don't count philosophical misconceptions, gross limitations on applicability, and blatant explanatory inconsistencies, then no established body of knowledge in ANY field has ever been overturned. On the other hand, if you DO count those things, then mathematics has been distinguished by almost continuously overturning its past forms and establishing new modes of thought and standards of rigor. For example, people had given proofs of the fundamental theorem of algebra before Gauss, but he showed that all his predecessors had been insufficiently rigorous and their proofs were unreliable.  His proof was then accepted as the first truly rigorous proof.  Today any freshman can tell you that Gauss's methods were insufficiently rigorous, and yet many professors will tell you that no mathematics has ever been overturned.  So it goes.


Surely the correct view of Cantor and his naive set theory is that, like Newton and his fluxions, his intuition was strong enough to compensate for a somewhat informal foundation.  All of which supports the point that mathematical knowledge is as likely to come from intuition as from formal axiomatic reasoning, and therefore it is of a kind with scientific thought.



Day Two:  Trial and Error


Simp:  I must say, Salviati, that your reference to computational time hierarchies (polynomial time, exponential, etc) astonishes me. Such time hierarchies only affect the practical issue of intractability, but this is precisely what mathematics is not concerned with.  "Easily" doesn't matter in mathematics.


Salv: The irreducible formal/computational distances between elements of a formal system are becoming more and more conspicuous, mainly due to the computer.  The old style of mathematics paid no attention to this structure, largely because it was too difficult to study in detail without computers.  But the interpretation (or modeling if you prefer) of formal mathematics as computation beginning (after Leibniz) with Godel, Turing, Post, et al, has shown that contemplation of the "computer model" of mathematics can yield very profound results.  I think it's natural to expect that the "intractable computational structure" of formal systems will likewise exert an influence on the way we view foundational questions.


Simp:  Well, in any case, I must object to your characterization of Euclid's Elements as "shot full of holes".  Surely no respectable mathematician shares your opinion.  Perhaps Euclid used some tacit assumptions which he overlooked to list, but nothing more serious than that.


Salv:  I'd refer you to any of several excellent books on the history and philosophy of mathematics.  For example, Michael Crowe remarks


...as early as 1892 C.S.Peirce... summarized a conclusion reached by most late-nineteenth-century mathematicians: “The truth is that Euclidean geometry, instead of being the perfection of human reasoning, is riddled with fallacies...”


The numerous deficiencies in Euclid as an axiomatic system are well-documented, and easy to see.  Of course, this doesn't reflect badly on Euclid because he wasn't in the business of creating axiomatic systems in the modern sense.  He understood himself to be engaged in what we would now call physics, i.e., saying things about spatial relations in the real world.  Read Heath's translation of The Elements, then read Aristotle on the structure of various fields of knowledge (including mathematics), and then have a look at Hilbert's modern axiomatization of geometry.


Simp:  You say Euclid was engaged in physics?!  Lines without breadth, points having no parts...  No perfect straight lines or circles in Nature...  In short, I don't think so.


Salv:  Idealization is an aspect of physics just as much as of mathematics.  There is no perfectly ideal gas, and yet physicists work out the implications of ideal relations such as pV=uRT.  This doesn't imply that physicists are uninterested in actual gases.  Similarly Euclid was idealizing real circles and lines, and working out the implications of those ideal forms, but his interest in those particular ideal forms was due to their correspondence with true spatial relations.  He wasn't interested in formal relations based on artificial or arbitrary axiom systems.  His axioms were specifically designed to reflect the "truth" about (ideal) spatial relations, just as the physicist establishes premises that reflect the truth about (ideal) gases.


Simp:  Perhaps so, but let's return to your claim that Gauss over-turned an established body of knowledge with his proof of the fundamental theorem.  A single faulty proof by D' Alembert is not exactly an "established body of knowledge".


Salv:  It wasn't just a single proof by D'Alembert.  Gauss began his doctoral dissertation in 1799 with a critique of the inadequate rigor in the purported "proofs" of the fundamental theorem given by D'Alembert (1746), Euler (1749), De Foncenet (1759) and Lagrange (1772) over the preceding half-century.  Ironically, as mentioned previously, Gauss's proof is now criticized for lack of rigor on very similar grounds.


Simp: I would have to be shown Gauss's lack of rigor.


Salv: In Part I of Section 19 of his dissertation Gauss says "but since the value at x is negative and at y is positive, it must equal zero somewhere between x and y."  Today this is known as using visual evidence, and is not accepted as valid proof because circumstances are known in which it doesn't work, i.e., it is not a reliable method of inference.


Simp:  It's a standard property of continuous functions (and of derivatives too); not visual evidence but valid argument.  Are you claiming that Gauss used it for something that wasn't, say, continuous? 


Salv: He used it for something that wasn't necessarily continuous, i.e., the continuity of the locus in question had not been established.  Ironically, it was precisely on these grounds that Gauss had criticized some of the prior proofs, but then he committed the same error (in another part of the proof) himself.


Sagredo:  Let me interrupt, Salviati.  Your the argument that our confidence in ZF rests on an "incomplete induction" is completely unconvincing.


Salv:  Perhaps you think the induction I'm referring to is limited to conscious induction at the level of the axiom system itself.  That's not the case.  The first and most important use of induction is in the formation of your clear and immediate notions of the sequence of natural numbers - your mental model of the naturals.  You may have begun to form those notions when you first opened your eyes.  Or, if you prefer, your common notions (to use Euclid's phrase) may be a consequence of the inherent wiring of your brain which evolved by a process of trial-and-error interactions with the environment - again involving incomplete induction, albeit unconscious on your part.  Or you may believe that your clear and simple notions arise from an entirely metaphysical source, i.e., entirely unconditioned by any external factors, which is a form of mysticism.  Or etc. etc.  It isn't my purpose to list all the possible theories for the sources of knowledge efficacious grace, merely sufficient grace,...) but simply to observe that there is no reason to believe mathematical knowledge at the fundamental level is essentially different from physical knowledge.


Sagr: Whatever "induction" that may be involved in forming a picture of the natural numbers is not at issue. The "incomplete induction" on which you claimed that our confidence in a formal system is "necessarily based" was, according to your own later explanation and quotation, that expressed in the reflection that "mathematicians have been working with ZF for most of this century, and no contradictions have been discovered yet".


Salv:  No, my claim has been and remains that induction is used throughout the establishment of our thought processes, from the formation of rudimentary notions to the development of more refined intuition to the construction of formal axiomatic systems.  Furthermore, at each of these levels there are historical examples illustrating the fallibility of our inductive judgments.


Everyone seems to agree that we arrive at our rudimentary notions, if not inductively, then at least by some means other than formal deductive reasoning.  (One can hardly disagree with this.)  Your original disagreement was with the idea that induction is also applied to assess the validity of high-level axiomatic systems.  In particular, you claimed that "no one's confidence in ZF is based on induction".  To show that your claim was false I presented a quote from a mathematical philosopher explicitly citing an inductive argument for the consistency of ZF. 


I could quote many others.  For example, when this was discussed on the internet last year one of the participants chided me for suggesting that mathematical knowledge is ultimately based on induction and that our confidence in results within ZF must always be less than perfect:


[This] suggests yet again that you don't know beans about what you are talking about.  A proof from ZF brings with it the supreme confidence that a century of working with ZF and beyond has given us.


In view of remarks like this it's hard to see how anyone can disagree that mathematicians routinely invoke inductive arguments to support the validity of axiomatic systems.  I also cited Frege as an example of an axiomatic system that survived the careful scrutiny of eminent mathematicians for several years and that failed only when confronted with an actual example of a logical inconsistency.


You go on to assert that "Whatever "induction" may be involved in forming a picture of the natural numbers is not at issue."  But since this in a discussion about the origins of mathematical knowledge, I simply have to disagree with you.  The formation of our rudimentary notions and intuitions is not only at issue, it is the issue.


Sagr: What I claim is that the actual behavior of mathematicians - as opposed to their occasional philosophical asides - does not support the view that they base their confidence in any formal system on such "incomplete induction".


Salv: You have this exactly backwards.  The actual behavior of most mathematicians is to rely heavily on the integrity of the mathematical community for assurance of the correctness of the vast body of mathematical work they could never personally hope to check in detail, often not even completely mastering results on which their own work is based, and certainly not worrying about foundational questions such as the ultimate consistency of the formal system in which they happen to be working.  It's only in their "philosophical asides" that most mathematicians are likely to overlook how they really do business, and claim that mathematical knowledge is certain and sure because we can see it in all it's irrefutable inevitability all the way to the ground.



Day Three: Seeing It


Simp: The view that we "see" N is held by just about everybody except strict formalists... it was in reply to the question: how do we come to believe Consis(PA) or (ZF) I said, by having a model of PA, or ZF.  People who believe  Consis(PA) do so because they see PA holds in N.


Salv:  So your claim is that we "see" N (the set of natural numbers) and then we "see" that PA holds in N.  As far as I can tell, you don't claim this "seeing" is a process of deductive reasoning. You're evidently content to simply name it "seeing" and declare that it can't be analyzed any further.  The only properties of "seeing" that you acknowledge are (1) it is the ultimate source and justification of mathematical knowledge and (2) it differs from other sources of knowledge in the indubitability of its results and its complete independence from experience.  My claim is that this "seeing" is ultimately derived from experience, mainly from our most primitive perceptions of consistency.  (I trust it goes without saying that the distinction between internal and external experience is itself a subjective judgment based on experience.)


Simp:  I'll ask again: do you consider N, +, *  clear and precise? 


Salv:  "Clear and precise" are terms that can be (and often are) applied to various notions in physics, so your question doesn't get at a potential difference between physical and mathematical knowledge. To be relevant to the discussion at hand you must have meant to ask a much sharper question, namely, whether N is perfectly clear, transcendentally clear, more clear than anything based on experience can be.  The answer is no, because N is, in fact, based on experience at various levels.  Of course, the complete set N is an extrapolation, because we don't actually have any direct experience of completed infinite sets, the adoption of which as a foundation of mathematical knowledge is a relatively recent event.  Completed infinities would have (and did) horrify the ancient Greek philosophers and mathematicians.  This doesn't prove that N is ill-founded, but it certainly casts doubt on your comfortable premise that "N" can claim the universal assent of all sentient beings.  Furthermore, the allegedly perfect correspondence between "N" and PA is far from clear, not least because N itself is an intuitive notion that may not even be perfectly capturable as an axiomatic system.


Simp:  Returning to your claim that "the intractable computational structure of formal systems will exert an influence on the way we view foundational questions", I agree that this is very possible, but not in the way you suggested -- tolerating contradictions if they are 'hard to reach'!   Note: One proof of a contradiction suffices; so input-length asymptotics, e.g. P, NP, Exponential time... hardly apply anyway.


Salv: You should finish your thought, Simplicio.  One proof of a contradiction suffices... to do what?  In the context of a discussion about whether ZF (for example) might be inconsistent, someone invariably raises the objection you have made, which can be summarized by the following argument for the consistency of ZF:


1. If ZF is inconsistent we would be able to prove 1 = 0 in ZF.

2. We are not able to prove 1 = 0 in ZF.

3. Therefore ZF is consistent.  QED.


The problem with this argument is that [1] is not necessarily true. There is a similar statement, let's call it [1'], that is true:


1'.  If ZF is inconsistent it would be possible to prove 1 = 0 in ZF.


Notice that I've changed  "we would be able"  into  "it would be possible".  This is a distinction that mathematicians have often been reluctant to make (cf Hilbert, "we shall know"), but the influence of computers and the increased focus on the reality of cognitive intractability is forcing mathematicians to view the nature of mathematical knowledge in a new light.  The complete argument based on [1'] becomes


1'. If ZF is inconsistent it would be possible to prove 1 = 0 in ZF.

2'. It is not possible to prove 1 = 0 in ZF.

3.  Therefore ZF is consistent.  QED.


In this form, item [1'] is okay, but we have no justification for asserting [2'].  All we can really claim to know is [2].  Thus, neither form of the argument is valid, because there is a difference between what could be done and what we can do.


Simp:  Let me return to an earlier point.  Cantor didn't misuse set theory... if the Encyclopedic Dictionary of Mathematics seems to indicate such a thing, then the wording is unfortunate, to say the least.


Salv:  Cantor's original axiomatization of set theory led to inconsistencies.  It's remarkable that in this instance you concede (in fact, insist) that a flawed formal foundation is irrelevant to the quality of the mathematics, whereas previously you asserted that a single implicit contradiction utterly invalidates every result derived within a given formal system.  Your view of Cantor's (and Euclid's) work reveals that you know this reductionist/formalist view is not justified.


Cantor himself warned others about the potential for paradoxes implicit in his original definitions and premises, once he recognized the problems himself.  In particular, his original definition of a "set" turned out to be insufficiently restrictive, as he himself said.  Thus, by his own admission, his original work was based on premises and definitions that were implicitly inconsistent.


Moreover, the flaw in the premises was not "seen" by thinking harder about the premises and whether they really do correspond to some "model" that we know for certain is consistent; it was revealed only by actually bumping into inconsistencies, and then working backwards to arrive at the conclusion that the premises must be flawed.  Then they dreamed up a new set of premises that would avoid those inconsistencies.  Moral: the consistency of our results is not due to the clarity and precision of our premises; rather we shape (and reshape) our premises to make the formal system give the results that we've already determined to be consistent by external means - just as we do in physics.


Simp:  Well, I would agree to the extent that usually development of a mathematical theory comes first, and axiomatization afterwards.


Salv:  Precisely, and this is the hierarchy conceptually as well as chronologically.  Math is developed informally, and then someone gets around to attempting to formalize it, to make it rigorous, thereby ostensibly setting it apart from informal sciences like physics.  But it usually happens that the early attempts at axiomatization are flawed, and we "see" it quickly.  Then we improve the axioms and they stand up a while longer, but we come across another flaw.  We fix the axioms again (see Proofs and Refutations, Lakotos), and so on, until we go for years without anyone finding a flaw.  Then we declare the project complete.  Clearly the governing force behind this project has not been formal axiomatic reasoning from manifestly clear first principles; rather the formal system has been shaped and corrected at every stage by informal reasoning.  That's why I began by saying that we're not justified in differentiating between math and physics knowledge on the grounds that math knowledge is based on formal deductive reasoning while physics knowledge is based on informal reasoning.  In truth, they are both based on informal reasoning from incomplete and (for all we know) flawed premises.  If and when the consequences of any flaws become apparent, we will revise our premises.


Simp:  It may be true that informal reasoning is often used in mathematics as well as physics, but it doesn't follow that informal mean non-rigorous.  For example, nobody carries out proofs as Predicate Calculus formal deductions.


Salv:  But surely the "fact" that no one practices formal proof doesn't imply that informal proof is rigorous.


Simp:  Let me take another approach to distinguishing between math and physics.  The big difference is, what physical sciences study is in the world; what math studies isn't.


Salv:  Here you're making a very conventional distinction between internal and external experience, and claiming that the former is not "in the world".  It isn't clear that more sophisticated theories in both math and physics will necessarily bifurcate our experience along these same lines.  Thus, on a purely experiential level, there is no reason to believe mathematical knowledge is essentially qualitatively different from physical knowledge.


Simp:  Well, studying N does seem different from studying things in the world, like accelerations, electric charges, temperatures... I don't see what argument you have made that it isn't.


Salv:  As Aristotle observed, in every field of knowledge there is an inductive and a deductive component.  I would say that your study of N "seems" (to you) to lack an inductive component only because you have never reflected on the ultimate source of your primitive notions about N.


Simp:  Granted that the foundations of knowledge in both mathematics and physics contain inductive components, I would say the crucial difference is that in mathematics the goal is deduction, whatever the path that takes us there; the end result has to be a correct proof.  The subject-matter is abstract; there is no 'agreement with experiment', no perihelion anomalies or anything.


Salv:  So now you're saying the difference between math and physics is not in what we do, but why we do it (i.e., what's the goal).  For example, you would say that Kepler's study of conics was physics, whereas Apollonius's study of conics was math.  Riemann's study of geometry was math but Einstein's was physics, and so on.  I would argue that in all of those cases the motivations had more in common than you might think.  Of course, we have Gauss's comment that geometry is actually a part of mechanics.  You might also have trouble classifying things like string theory.  Some parts of modern physics have little or no connection with the kind of "experiment" you have in mind.  On the other hand, in the broader sense of experience, both math and physics are inextricably bound up in it.


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