Ayer writes in “The A Priori” that “The principles of mathematics and logic are true universally simply because we never allow them to be anything else.” Compare this position to Hempel’s in ‘The Nature of Mathematical Truth". Would Ayer care about the success or failure of the technical program like Frege’s or his successors’?

Ayer vs Hempel: Math and Truth

If you put a gun to the head of Alfred Jules Ayer and demanded he describe the difference between a mathematician and a linguistician, I would be unsurprised if he cited the difference to be their bathing patterns (a valid critique, shareable by any who finds themselves inside Fine Hall). To Ayer, the mathematician is simply a specialized linguist, concerned solely with synonyms and equivalent definitions. Before I write on the comparison between Hempel and Ayer’s positions, I’ll talk a bit more explicitly on what Ayer’s position is in the first place. After that, I’ll detail why I don’t think Ayer would care very much about Frege’s program or any other similar program for that matter.
The prompt’s quote is a good example of Ayer’s position on truth in mathematics, but the latter half of the quote is particularly important: “the principles of mathematics and logic are true universally simple because we never allow them to be anything else. And the reason why we never allow them to be anything else is that they are purely linguistic.” (Ayer, “Language, Truth, and Logic", p. 71) He explains that these linguistic statements are disjoint from the empirical world, that logic and math “do not describe the empirical world, but simply record our determination to use symbols in a certain fashion.” (Ayer, “Language, Truth, and Logic", p. 80) To Ayer, mathematics is a sequence of tautologies that comes from linguistic methods following the selection of a system (presumably via axioms, or as Ayer says “definitions”). In an example of one of these tautology sequences, Ayer would say that in the equation “2+2=4”, “2+2” and “4” are linguistically equivalent statements being translated between. The tautology being made here is that in our choice of defining addition, numbers, and equality (the definitions), all of the “words” have already been selected. All that is left is for translations between these statements into other, synonymous statements.
In some respects, Carl Gustav Hempel is in agreement with Ayer. They agree, for example, on the analytic and a priori nature of mathematics. “The statement that 3+2 = 5, then, is true for similar reasons as, say, the assertion that no sexagenarian is 45 years of age. Both are true simply by virtue of definitions or of similar stipulations which determine the meaning of the key terms involved.” (Hempel, “Nature of Math Truth”, p. 544) In essence, statements in mathematics are automatically true so long that they are internally consistent with previous definitions or statements. Ayer has a similar example, but with “7 + 5= 12”, (Ayer, “Language, Truth, and Logic", p. 73) I suppose to prove to any future detractors that he can indeed count past ten. They both seek to disconnect empirical science from math, with similar statements from both. From Hempel: “while the statements of empirical science, which are synthetic and can be validated only a posteriori, are constantly subject to revision in the light of new evidence, the truth of an analytic statement can be established definitely, once and for all.” (Hempel, “Nature of Math Truth”, p. 544) Ayer agrees: “these analytic propositions are necessary and certain. We saw that the reason they cannot be confuted in experience is that they do not make any assertion about the empirical world.” (Ayer, “Language, Truth, and Logic", p. 80) Mathematics, they argue, is truth– truth that cannot be found in the empirical world, where statements constantly need to be adjusted to fit new data. Similarly, they agree that the principles of mathematics are “devoid of all factual content; they convey no information whatever on any empirical subject matter.” (Hempel, “Nature of Math Truth”, p. 552) No matter the mathematical statement, it conveys no information in and of itself about the world. That is not at all equivalent to Ayer and Hempel saying that mathematics cannot be applied, or that it cannot be somehow useful, but that in order for mathematics to describe the world, it must be applied – upon which it would no longer be a “principle of mathematics.”
Extending upon this, Ayer has an example of “a being whose intellect was infinitely powerful would take no interest in logic and mathematics. For he would be able to see at a glance everything that his definitions implied, and, accordingly, could never learn anything from logical inference which he was not fully conscious of already.” (Ayer, “Language, Truth, and Logic", p. 82) The logical inference in question is important to be noted as of linguistic nature. The being of “infinite intellect” has an infinite dictionary filled with infinitely equivalent definitions. In a similarly important example, Hempel speaks on mathematics as the “theoretical juice extractor:” “in the establishment of empirical knowledge, mathematics (as well as logic) has, so to speak, the function of a theoretical juice extractor: the techniques of mathematical and logical theory can produce no more juice of factual information than is contained in the assumptions to which they are applied.” (Hempel, “Nature of Math Truth”, p. 553) In this analogy, the system acts as a fruit to be juiced by a mathematician, like Euclid juicing the Pythagorean theorem from the fruit of Euclidean geometry. It is in these examples, however, that the differences between the two are found. Where Ayer focuses on the linguistic nature of mathematics, Hempel focuses on how the setup of axioms or “definitions” and the overall system results in mathematical truth. The techniques of “mathematical and logical theory” are how the “juice” gets extracted– not an infinite dictionary. While in both cases no new information is created – synonyms vs juice, both the words and orange existed before – the manner in which the information changes states is quite different.
Hempel rigorously details how the changing of states, arisen via deduction and system analysis, elucidates interesting properties depending on the choices of the system. Where Ayer does not focus much at all on the nuances of selecting individual systems, for Hempel the systems are in fact exactly what is so important about mathematics. Where to Ayer, the mathematician, logistician, and linguistician are isomorphic – simply translating definitions into new forms – Hempel stakes out a claim for what the mathematician and logistician actually do: deducing and extracting meaning from systems. In the example of the “juice extractor” analogy, Hempel discusses the Peano axioms – fruit – and the resulting mathematics – juice. What is true for Peano Arithmetic is true within that system. Where Ayer would describe all mathematics, including Peano Arithmetic, as true by definition, for Hempel truth in the context of a system like Peano Arithmetic is true within the framework of the axioms. Where Ayer finds the mathematician to be simply translating between synonyms, Hempel believes that the process of “juice extraction,” exploring the systemic implications of the axioms, uncovering non-trivial truths through deduction, is entirely the point of mathematics.
It is with this in mind that Hempel’s interest in Frege’s program makes so much sense: Frege’s logicism was an attempt to provide a fundamental system within which the entire field of mathematics can be derived. To Hempel, this is a hugely important undertaking, the selection of the “fruit,” as it were– the systemic foundations for the entire field of mathematics. Frege offered upon Hempel a system to study, and one with wonderfully sweeping consequences for all of mathematics. With respect to Ayer, however, frankly speaking I do not believe Ayer would care in the slightest about Frege’s program, or any other similar program. For Ayer, all of it is the same– just different synonyms, and all equivalent truths. No matter the technological complexity of the project, a tautology-machine is nothing but a tautology-machine, and to him does not affect or detract from his point whatsoever. Where Hempel’s primary focus is on systems and the selection process of those systems, Ayer has no focus or interest in the nuances of those systems.
While Ayer and Hempel share similar ideas on truth in mathematics– the nuance between them is critically important to understanding either of their ideas. Both agree on the analytic and a priori nature of mathematics– 7 + 5 is 12, and so too 3 + 2 is 5. They both agree that it is necessary to separate the empirical world from mathematics, that what is applicable to the real world from mathematics is nothing more than a useful coincidence. Ayer and Hempel agree that the principles of mathematics are the stevia of science, an artificial sweetener that contains zero factual knowledge, and that it is in the application of coincidentally useful mathematics that can produce actual knowledge (like Euclidean geometry). Ayer’s concept of mathematics as a specialized subset of linguistics, which – while justifiable – ignores the nuances of the specific systems and axiom selections that go into producing mathematical thought. Hempel’s focus on specific systems gives a more descriptive (and dare I say it, useful) picture of mathematics. Because of Ayer’s disinterest in systems, I believe that Ayer would be similarly disinterested in technical programs like Frege or any of his successors.

Bibliography
Ayer, Alfred Jules (1936). Language, truth and logic. London,: V. Gollancz.
Hempel, Carl G. (1964). On the Nature of Mathematical Truth. In P. Benacerraf H. Putnam (ed.), Philosophy of Mathematics. Prentice-Hall. pp. 366–81.