True or False? A Case in the Study of Harmonic Functions

Abstract

Recent mathematical results, obtained by the author, in collaboration with Alexander Stokolos, Olof Svensson, and Tomasz Weiss, in the study of harmonic functions, have prompted the following reflections, intertwined with views on some turning points in the history of mathematics and accompanied by an interpretive key that could perhaps shed some light on other aspects of (the development of) mathematics.

Appendices

Appendix 1: Frequently Asked Questions

Specialization has its side effects, sometimes amusing.⁴⁸ The typical first move of the editors of the journals to which we first submitted our paper was to send the paper to an expert in logic as well as to an expert in analysis. And then the expert in logic suggested to consult an analyst, and the expert in analysis to consult a logician.

We find it useful to relate the most significant reactions and questions prompted by these results, that were registered when the paper was submitted for publication or presented at conferences (together with our replies). Why useful? Because most of them reveal philosophical preconceptions. Some questions overlap, to some extent, but we find this insistence also meaningful.

• (1) Q. Perhaps (SSS) has something to do with issues of cardinalities?

  A. Not quite. If we assume CH, then (SSS) is resolved in a definite way. However, if we do not assume CH, then the answer will still depend on the particular model of ZFC we are adopting.

• (2) Q. What happens in our usual model of ZFC?

  A. There is no such a thing as “our usual model of ZFC”. As a matter of fact, we do not even know whether there is a model of ZFC, because we do not know whether ZFC is consistent or not. Mathematicians live dangerous lives, without even realizing it.

• (3) Q. Your models of ZFC must be pretty wild!

  A. No, they are not. They contain all the usual object of mathematical analysis. In them 2 + 2 = 4, there are infinitely many prime numbers, etc. None of them is “wild” in any sensible sense.

• (4) Q. In retrospect, it is not difficult to see that this had to end this way.

  A. Well, a posteriori it is easy to rationalize other people’s work. Bacon’s words apply here as well, as far as we know.

• (5) Q. I also have asked advice from an expert in logic and set theory. He points out that it would suffice to have a model of ZFC in which (1) property D, and (2) Unif(W) = c holds.

  A. This observation is indeed already contained in our paper.

• (6) Q. (The expert in logic points out that) It should be made clear that both [(1) and (2) in Question (5)] follow from Martin’s axiom.

  A. We did not insist on Martin’s Axiom in order to avoid distracting those readers who do not have an extensive background in modern logic.

• (7) Q. Although the results in this paper provide a somewhat surprising answer to issues raised by Rudin, I suspect that it would be of interest primarily to logicians rather than analysts.

  First answer. Well, problems do not come up with a label attached to it. We have chosen a problem, not the methods that have led us to its solution. Indeed, it took us a long time to find the solution because we were looking at the wrong kind of answer.

  Second answer. Your point of view seems to indicate that, if a problem in mathematical analysis cannot be settled within ZFC, then it is not of interest for analysts, but this positions appears dubious in itself, for it appears to imply that mathematical analysis lives within ZFC. Mathematical analysis thrived well before Zermelo was born (1871). We are not claiming that the formalism of set theory did not subsequently help the growth of mathematical analysis, as well as of other areas of mathematics. However, most working analysts do not possess anything more than a vague notion of what the ZFC axioms are, and this fact, in itself, should disprove the position that is implicit in the question.

  Third answer. It is not the case that we proved a result in logic because we were not smart enough to prove a result in analysis. We addressed a question in analysis (that came up in a paper by Littlewood in 1927, and in one by Rudin in 1979), and we proved that the answer could not be given within the borders that are traced by the ZFC axioms.

• (8) Q. The analytic parts of the arguments use fairly standard techniques.

  A. This comment overlooks the fact that the solution is based on the combination of methods coming from different areas, and an important ingredient in the proof is an insight about the place where a connection with logic is possible. We had to break the problem down into its parts or elements, from the point of view of mathematical analysis, but nobody knew a priori how to perform this analysis. Indeed, there is definitely not a unique way to decompose an object into different parts: It can be done in different ways, and we had to do it in the right way. A mental experiment will perhaps clarify this point. Suppose you have been working on a problem on elliptic curves for 10 years, and at some point you ask an expert in logic for help, in order to show that the statement encoding the problem is independent of ZFC. The expert in logic will not be able to help you immediately, because it would not be clear where exactly the tools from logic should be applied. They cannot be applied at any place of the problem chosen at random. One first needs insight about the place where logic could perhaps be applied.

• (9) Q. He (an expert in logic) also comments that a certain remark⁴⁹ in your paper seems to confuse truth with derivability.

  A. Of course. A sentence should not be detached from the context and much less from the one immediately following it. That remark was meant to introduce to these issues the average reader, who most likely does confuse truth with derivability. So, as a rhetorical device, we address a hypothetical reader who confuses truth with derivability, and proceed to explain the difference, in the parts that follow that remark. Indeed, right after that remark we write: “However, Gödel’s work warns us of other possibilities”, and then we proceed to clarify the issues and show the reader that the answer may elude the Law of the Excluded Middle.

• (10) Q. Since an approach is a fairly arbitrary subset of the Cartesian product of the boundary of D with D, in retrospect your result can be rationalized.

  A. Other examples in analysis show that this rationalization is not a priori infallible, as we have seen. Therefore, the rationalization is only an exercise in hindsight.

• (11) Q. This is not a result in mathematical analysis!

  A. The problem came up within mathematical analysis, first in a paper by Littlewood in 1927, then in a paper by Rudin in 1979. We have chosen the problem. We could not chose its solution.

Appendix 2: A Matter of Choice. II

Let us go back to the attack against Zermelo. As observed by Moore (Moore 1983, pp. 131–132), Borel, “in his monograph, unbeknownst to himself, had repeatedly violated his own philosophical dictum by implicitly using the Axiom of Choice at several junctures”. “In a note at the end of his book […] he claimed that the set of all Borel sets has the cardinality of the continuum. Much later, however, mathematicians discovered that if the Axiom of Choice were false, it could happen that the set [of all Borel sets] has power greater than [that of the continuum]” (ibidem, pp. 132–133). Baire has also implicitly used AC in his work, and “If [AC] were false, then all of Baire’s results […] would be false” (ibidem, p. 135). As for Lebesgue, he was faced with a dilemma. “For, if [AC] were false, it could happen that Lebesgue measure is not countably additive” (ibidem, p. 137). Indeed, without AC, it is not possible to show that the set of all real numbers is not equal to the countable union of countable sets (ibidem, p. 133). The goal of the research in Solovay (1970) was to show that AC is necessary to show the existence of sets of reals that are not Lebesgue-measurable. He proved that, if the existence of inaccessible cardinals is consistent with ZF,⁵⁰ then there is a model of ZF where a weak form of AC holds, and where each set is Lebesgue measurable. He writes that “of course the axiom of choice is true, and so there are non-measurable sets”.

Appendix 3: A Tale of two Hemispheres

We would like to expand a bit Hadamard’s observation, that really essential progress in mathematics has been achieved by annexational moves that incorporated, within mathematical territory, notions that were previously considered to lie outside its realm.

Why “outside its realm”? Either because it was previously considered impossible to talk about them (because the mathematical language of the time was not rich enough) or because those notions had no sensible “meaning”. It thus seems that these annexational moves can be grouped into two categories. In the first, (the expressive power of) mathematical language is enriched, in such a way that it becomes possible to talk about the new mathematical objects that are somehow known to have a “geometric” or “objective” existence. In the second, one finds concrete (usually visual or geometric) representations (or models) for notions that have surfaced from language but that were otherwise apparently meaningless. It seems to us that it would be interesting to test-drive this classification by looking at the numerous annexational moves that have occurred in the history of mathematics. We will briefly review just three cases.

An example in the second category is given by the discovery of i, the square root of −1, a number whose square equals −1. This notion popped up from language, from the algebraic language that was developed to solve algebraic equations. This notion was useful to solve those equations, but for about two centuries nobody could attach to it any meaning. Its existence as a number was in doubt, for it could be neither a positive number, nor a negative one (since its square would not be negative, in any case). Mathematicians were puzzled by these “numbers”, formed by taking the square root of negative numbers, and referred to them as “quantitates surdae ac ignotae” (where “surdae” stands for “absurdus”). (Mathematical) language has a tendency to grow on its own, and it took not too long for treatises dedicated to calculations with these numbers to be written, but their mistery was dissipated only when Argand and Gauss found for them a simple geometric representation, or model, as points in the plane, a model large enough to host the old, familiar bona fine (positive and negative) numbers as well. A trace of the ancient mistery has remained in their name: imaginary numbers.

Another example in the second category is given by the so-called distributions, among which the most famous appears to be the “Dirac’s delta function”: a “function”, denoted by δ, defined by formal properties that actually made it impossible for such a function to exist, and yet a convenient tool in many calculations in different areas of mathematics, engineering, and physics. This puzzle was resolved in a way that has several points in common with the previous example: these objects were “represented” in a “space” that was large enough to host the old, familiar, bona fine functions as well.

An example in the first category is given by the discovery of incommensurable quantities, which

was taken to imply that geometry had a larger scope than arithmetic—there was simply no number, one thought, that could express the relation between the side and the diagonal of a square, which on the other hand everyone could see. This larger scope, hence also higher status, of geometry was one main reason why to prove something mathematically, for a long time, was to prove it ‘the way the geometers do it’: more geometrico.⁵¹

The impasse produced by the discovery of incommensurable quantities on Greek mathematicians was of a linguistic nature (their numerical language was unable to express geometric reality) as well as of a philosophical nature (their numerical language mirrored an erroneous belief regarding the nature of things). The process, that led Greek mathematicians to express a solution to the impasse in the geometric terms of the theory of proportions, is a good illustration of the dictum “Those who cannot say what they think are confined to think what they can say”. Kurt von Fritz has written a fascinating analysis of that process.

The Greeks had two terms for ‘word’: epos and logos. Epos means the spoken word, or the word which appeals to the imagination and evokes a picture of things or events. This is the reason why it is also specifically applied to epic poetry. Logos designates a word or combination of words in as much as they convey a meaning or insight into something. It is this connotation of the term logos which made it possible for it in later times to acquire the meaning of an intrinsic law or the law governing the whole world. If logos, then, is the term used for a mathematical ratio, this points to the idea that the ratio gives an insight into a thing or expresses its intrinsic nature. In the case of musical harmonies the harmony itself would be perceived by the ear, but it was the mathematical ratio which, in the mind of the Pythagoreans, seemed to reveal the nature of the harmony, because through it the harmony could be both defined and reproduced in different media. […] Logos or ratio meant the expression of the essence of a thing by a [pair] of integers. It had been assumed that the essence of anything could be expressed that way. Now it had been discovered that there were things which had no logos. When we speak of irrationality we mean merely a special quality of certain magnitudes in their relation to one another […] but when the Greeks used the term alogos they meant originally […] that there was no logos or ratio. Yet this fact must have been very puzzling. It had been generally assumed that two triangles which were similar, i.e., which had the same ornamental appearance, though differing in size, had the same logos, i.e., that they could be expressed by the same set of integers. In fact, this is clearly the original meaning of the term ho autos logos (the same logos), which we translate by “proportion”. But two isosceles right-angled triangles had still the same ornamental appearance, and therefore should have had the same logos. In fact, it seemed evident that their sides did have the same quantitative relation to one another. Yet they had no logos.⁵² Very soon they […] established a criterion by which in certain cases it can be determined whether two pairs of incommensurables (which in the old sense have no logos at all) have the same logos. The terminological difficulty created by this seeming contradiction in terms is reflected by the fact that for some time the term alogos for irrational was replaced by the term arrhetos (inexpressable) […] First the term rhetos (rational) is created in contrast to arrhetos. Then the term arrhetos disappears […] Finally, when logos had become a technical term and the incongruity of the statement that two pairs of alogoi have the same logos was no longer felt, the Greek mathematicians returned to the old terminology and called all irrationals alogos. ⁵³

The theory of proportions was not very efficient at a linguistic level. A language that proved more manageable grew out of the development of an efficient algebraic formalism, which ultimately made possible the arithmetization of analysis.

The proposed classification appears to reflect the opposition discrete/continuous, where in the first pole are located logic and language (logos), and in the second one our faculty for geometric, spatial representation (epos). The following excerpt shows that this opposition (together with various attempts to strike a balance, reach a synthesis, or formulate a unification) can be felt in the realm of literary creation as well.⁵⁴

Mi suerte es lo que suele denominarse poesía intelectual. La palabra es casi un oximoron; el intelecto (la vigilia) piensa por medio de abstracciones, la poesía (el sueño), por medio de imágenes, de mitos o de fábulas. La poesía intelectual debe entretejer gratamente esos dos procesos. Así lo hace Platón en sus diálogos; así lo hace también Francis Bacon, en su enumeración de los ídolos de la tribu, del mercado, de la caverna y del teatro. […] Admirable ejemplo de una poesía puramente verbal es la siguiente strofa de Jaime Freyre:

• Peregrina paloma imaginaria

• Que enardeces los últimos amores;

• alma de luz, de música y de flores,

• peregrina paloma imaginaria.

No quiere decir nada y a la manera de la música dice todo. Ejemplo de poesía intelectual es aquella silva de Luis de León, que Poe sabía de memoria:

• Vivir quiero conmigo,

• gozar quiero del bien que debo al Cielo,

• a solas, sin testigo,

• libre de amor, de celo,

• de odio, de esperanza, de recelo.

No hay una sola imagen. No hay una sola hermosa palabra, con la excepción dudosa de testigo, que no sea una abstracción. Estas páginas buscan, no sin incertidumbre, una vía media.