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I am reading Sieg's "Hilbert's programs and beyond" and I am having difficulty understanding this quote by Hilbert on page 74:

In the Paris Lecture Hilbert re-emphasized and expanded this point by saying: In the case before us, where we are concerned with the axioms of real numbers in arithmetic, the proof of the consistency of the axioms is at the same time the proof of the mathematical existence of the complete system [“complete system” is Ewald’s translation for “Inbegriff ”, WS] of real numbers or of the continuum. Indeed, when the proof for the consistency of the axioms shall be fully accomplished, the doubts, which have been expressed occasionally as to the existence of the complete system of real numbers, will become totally groundless.

Sieg continues with this:

He proposed then to extend this approach to the Cantorian Alephs. Consequently and more generally, the consistency of appropriate axiom systems was to guarantee the existence of sets or, as he put it in 1905, to provide an objective criterion for the consistency of multiplicities.

This is a common theme this book alludes to in Hilbert's early period, before 1920s. Hilbert believed that consistency of an axiomatic system implies the mathematical existence of the model satisfying those axioms. But it seems to me that this is only a consequence of completeness of the system in question and not a general fact at all. Namely, if some A is a syntactically consistent statement, assume that A is not true in any model. Then by completeness we have that ~A, which is true in every model, must be a theorem which contradicts the fact that A is consistent. So an existence of a model is a consequence of a consistent axiomatic system, but after the assumption of completeness, which I don't see mentioned in this book. This leads me to believe that maybe I am gravely misunderstanding the way these words were used in Hilbert's time, or that there might be some other thing I am missing. Why does Hilbert believe that consistency of axioms for real numbers would provide him with existence of the reals?

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  • $\begingroup$ But the real number system is already constructed from Peano arithmetic by Dedekind cuts or by Cauchy sequences by that time. Instead, the questions are about whether the systems so constructed are consistent or not, complete or not. $\endgroup$ Jul 18, 2023 at 5:11
  • $\begingroup$ @naturallyInconsistent No, Hilbert was against the genetic method of constructing the real numbers and proposed an axiomatic method of doing so. He took the axioms of an ordered field, archimedean axiom and added to it a metamathematical axiom of completeness. He then claimed that the consistency of this system would imply the mathematical existence of real numbers. That was the case before 1905 at least. This question is about why consistency would imply that existence. $\endgroup$
    – Anon
    Jul 18, 2023 at 6:35
  • $\begingroup$ Hmm, sounds like a rather profound question. Good luck $\endgroup$ Jul 18, 2023 at 8:01
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    $\begingroup$ I can quote this article Enrico Moriconi, On the Meanig of Hilbert's Consistency Problem, (2003), that seems to address exactly your question, and says that the link between consistency and existence must be found not in the completeness of second order logic, but in the completeness axiom, in the sense of completness of a number system. $\endgroup$ Jul 18, 2023 at 20:35
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    $\begingroup$ @TannerSwett He doesn’t argue that A is true in every model of T, just in some model of T. $\endgroup$ Jul 19, 2023 at 19:48

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Hilbert wrote this observation some decades before the formal theories of completeness (and incompleteness) took shape, so you can't expect him to be overly precise with regard to such notions. The main novelty of Hilbert's remark is precisely this: consistency is sufficient to ensure mathematical existence, and there is no need to argue that the geometric line is present in physical space somewhere or somehow. This may have been a novelty to his contemporaries but truth be told the observation is already in Leibniz, who spoke of imaginary roots, infinitesimals, and negative numbers as useful fictions, even though they may seem impossible in the physical world (which Leibniz referred to as the "phenomenal realm"). For further details on Leibniz's fictionalism see this recent article in British Journal for History of Mathematics. Of course, much of Leibniz's writing was in manuscript form until recently and was not necessarily available to Hilbert's contemporaries.

There is an insightful recent article by Rowe on Cantor and Hilbert; see particularly page 4, second column.

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I just want to add some context to Katz's very nice answer. Hilbert's work on foundations occurred in the aftermath of the intuitionistic criticisms by Brouwer and Weyl. (Weyl in particular must have upset Hilbert, as he was Hilbert's student.) They maintained that only constructive proofs of existence counted; "proving" existence by contradiction proved nothing, or so they claimed. Hilbert responded (from Reid's biography of Hilbert):

Taking the Principle of the Excluded Middle from the mathematician is the same as ... prohibiting the boxer the use of his fists.

Reid quotes an exchange between Pólya and Weyl:

"Hermann, that is mathematics in shirt-sleeves" Pólya told him, in other words, not completely attired. Weyl promptly offered to wager Pólya on the future of two specific propositions which would be eliminated from mathematics if Brouwer's ideas were to be accepted, as Weyl was convinced they would be---and within 20 years. The winner of the wager was to be decided by whether in 1938 Pólya was willing to admit that the following two propositions---

  1. That each [non-empty] bounded set of real numbers has a precise upper bound,
  2. That each infinite set of real numbers has a countable sub-set

---were in fact completely vague "and one could ask of their truth or falsity as little as he could ask of the truth or falsity of the main ideas of Hegel's philosophy'"

Cantor's remark, "The essence of mathematics is its freedom" is often quoted, but the context here are the attacks upon his work by Kronecker, a forerunner of the intuitionists. Dauben's biography of Cantor describes it this way:

For mathematicians, only one test was necessary: once the elements of any mathematical theory were seen to be consistent, then they were mathematically acceptable. Nothing more was required:

and then he quotes Cantor directly:

In particular, in introducing new numbers, mathematics is only obliged to give definitions of them ... As soon as a number satisfies all these conditions, it can and must be regarded as existent and real in mathematics... [Grundlagen einer allgemeinen Mannigfaltigkeitslehre]

This was in defense of Cantor's transfinite numbers. Recall here Hilbert's call-to-arms, "No one shall expel us from the Paradise that Cantor has created for us." This appeared in Hilbert's address, "On the Infinite", where he also said:

Also old objections which we supposed long abandoned still reappear in different forms. For example, the following recently appeared: Although it may be possible to introduce a concept without risk, i.e., without getting contradictions, and even though one can prove that its introduction causes no contradictions to arise, still the introduction of the concept is not thereby justified. Is this not exactly the same objection which was once brought against complex-imaginary numbers when it was said: "True, their use doesn't lead to contradictions. Nevertheless their introduction is unwarranted, for imaginary magnitudes do not exist."? If, apart from proving consistency, the question of the justification of a measure is to have any meaning, it can consist only in acertaining whether the measure is accomplished by commensurate success. Such success is in fact essential, for in mathematics as elsewhere success is the supreme court to whose decisions everyone submits.

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  • $\begingroup$ As the question was crossposted to math.stackexchange, I am also crossposting this answer. $\endgroup$ Jul 18, 2023 at 16:59
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Think back to when you learned linear algebra: \begin{align} x + y + z & = 2 \\ 6x - 4y + 5z & = 31 \\ 5x + 2y + 2z & = 13 \end{align} Does a solution to this system exist? (In this case, yes.) If no solution exists, then applying elementary row operations, you will at some point get $$ 0x + 0y + 0z = 1. $$ Otherwise a solution exists. Extrapolating from this, one could say that if one can never deduce a contradiction from the assumption of the existence of some mathematical object, then it exists. It was only decades later that Gödel showed that that is wrong, and Hilbert was not at all pleased by Gödel's discovery.

Gödel proved his two famous incompleteness theorems. He also proved what is known as Gödel's completeness theorem, which says that if no contradiction can be deduced from a set of first-order formulas, then a model of those formulas exists. I.e. every such formula with no free variables has either a proof or a counterexample.

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There is a historical context involved arising from the influence of Kant.

There had been a number of reasons for anti-Kantian backlash in the nineteenth century. However, his wider influence led to the notion that mathematics has been intimately bound with astrophysics. In a certain sense, mathematicians had been at risk of losing an independent identity.

The mathematical congresses began to be called, and authors like Cantor had begun to reassert Leibnizian principles (Kant had associated the principle of the identity of indiscernibles with logic and noumena in the same passage where he associated the separation of geometric points with mathematics and phenomena).

Relative to this, it is important to understand that Hilbert actually had two foundational periods. His "Foundations of Geometry" had been written in the early period. Apparently, he had some concern about the applicability of mathematics across all application disciplines. I believe this to be the origin of his formalism. And, this is compatible with the direction of establishing an independent identity for mathematics.

So, the question then arises, what constitutes existence for a formal mathematics?

I have never had the impression that Hilbert had shared the anti-Kantian view commonly promoted in foundations. He includes non-essential quotes from Kant in "Foundations of Geometry" and makes a nod toward Kant after introducing metamathematics (Kant's schema of number is characterized with concatenated strokes just like Hilbert's metamathematical numerals).

In "Critique of Pure Reason," Kant declares that logic only provides a negative criterion of truth (rejection of contradiction). It is reasonable to conjecture that this had had influence on Hilbert through German idealism even if he had never been aware of the passage directly.

As for later Hilbert, the introductory remarks from the "Foundations of Mathematics" continue to reflect a division between logic and mathematics. This may have been forced upon him by intuitionist criticism. And, his later work had been influenced because of "Principia Mathematica" by Whitehead and Russell. Nevertheless, Hilbert had not adopted logicism. There is no reason to interpret Hilbert in the context of modern analytic philosophy.

In any case, he expressly describes the consistency question as a logical problem to be addressed through formal axiomatics. He does not assert that mathematics is solely formal axiomatics. In the same passage he describes formal axiomatics as relying upon interpreting universal quantification relative to decidable equality. And, he acknowledges that formal axiomatics deprecates the existential quantifier. Moreover, formal axiomatics is not independent of other mathematics. Rather, mathematical practice outside of formal axiomatics ought to inform formal axiomatics.

From my reading, Hilbert changed his view across his career, although existence based upon non-contradictoriness had been retained for formal axiomatics because of his original motivations.

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    $\begingroup$ Thanks for an interesting post. But how does Kant come in here? Neither the question nor any of the other answers mention Kant. Which position are you arguing against? I realize you can't contribute comments unless your reputation goes up, but you could modify your answer to clarify this point. $\endgroup$ Jul 19, 2023 at 12:18
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    $\begingroup$ "authors like Cantor had begun to reassert Leibnizian principles": it is not clear what you mean by that. It is well known to Leibniz scholars that Leibniz adhered to the part-whole principle (a whole is greater than its part) and consequently rejected the notion of "infinite wholes" as contradictory (and therefore of no use in mathematics). Cantor, of course, went in precisely the opposite direction. So in what sense did Cantor "reassert Leibnizian principles"? $\endgroup$ Jul 20, 2023 at 6:56
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@Dr. Katz

The original poster concludes with the question:

"Why does Hilbert believe that consistency of axioms for real numbers would provide him with existence of the reals?"

So, the issue lies with Hilbert's views on mathematical existence and not with modern jargon highly influenced by analytical philosophy. I no longer have my pseudobiography on Hilbert, or I would try to be more specific concerning the origins of his formalist views. But, the question asks about opinions he held early in his career -- presumably before the publication of "Foundations of Geometry."

The entry,

math.stackexchange.com

/questions/56603/provenance-of-hilbert-quote-on-table-chair-beer-mug#56605

documents the commonly known quote associated with his early views.

As noted before, the literature appears to indicate that Hilbert never fully conceded to the disparagement of Kant pursued by others. It is in Kant where logic is claimed to be distinct from mathematics. And, it is difficult to believe that a man who quoted Kant in his publications had not been aware of his actual writings.

After speaking to mathematics in the Transcendental Aesthetic, Kant turns to logic in the Transcendental Logic. At approximately A60 and B84, Kant writes,

"But, on the other hand, as regards knowledge in respect of its mere form (leaving aside all content), it is evident that logic, in so far as it expounds the universal and necessary rules of the understanding, must in these rules furnish criteria of truth. Whatever contradicts these rules is false."

So, the role of non-contradictoriness is established. The paragraph of the passage above concludes:

"The purely logical criterion of truth, namely, the agreement of knowledge with the general and formal laws of the understanding and reason, is a conditio sine qua non, and is therefore the negative condition of all truth. But further than this logic cannot go. It has no test for the discovery of such error as concerns not the form but the content."

What I do definitely recall from the pseudobiography I once owned is that Hilbert's emphasis on consistency arose from his desire to locate any alleged inconsistency associated with mathematics to application domains and not mathematics itself. Kant's description of the limitations on logic seem to be precisely what Hilbert had needed.

I did try to be cautious since I have no direct evidence that Hilbert had been aware of Kant's actual statements.

As for Cantor, Footnote 6 of his Grundlagen contains the statement:

"The same epistemological principle can also be found in the philosophy of Leibniz."

Cantor is referring to a passage from Spinoza,

"By 'adequate idea' I understand an idea which, to the extent that it is considered in se, without relation to an object, has all the properties or intrinsic denominations of a true idea."

What immediately follows Cantor's reference to Leibniz is a rejection of the empiricist program of grounding knowledge upon subjective experience. Cantor writes:

"But, in my opinion these elements do not furnish us with any secure knowledge. For this can be obtained only from concepts and ideas that are stimulated by external experience, and are essentially formed by inner induction and deduction as something that, as it were, was already in us and is merely awakened and brought to consciousness."

The next footnote provides the comparison with Leibniz:

"The procedure in the correct formation of concepts is in my opinion everywhere the same. One posits a thing with properties that at the outset is nothing other than a name or sign A, and then in an orderly fashion gives it different, or even infinitely many, intelligible predicates whose meaning is known on the basis of ideas that are already at hand, in particular to related concepts. If one has reached the end of this process, then one has met all the preconditions for awakening the concept A which slumbered inside us, and it comes into being accompanied by the intrasubjective reality which is all that can be demanded of a concept; to determine its transient meaning is then a matter for metaphysics."

The stepwise refinement described by Cantor is to be compared with what Leibniz writes in "Elements of a Calculus":

"Two terms which contain each other but do not coincide are commonly called 'genus' and 'species'. These, in so far as they compose concepts or terms (which is how I regard them here) differ as part and whole, in such a way that the concept of the genus is a part and that of the species is a whole, since it is composed of genus and differentia. For example, the concept of gold and the concept of metal differ as part and whole; for in the concept of gold there is contained the concept of metal and something else -- e.g., the concept of the heaviest of metals. Consequently, the concept of gold is greater than the concept of metal.

"The Scholastics speak differently; for they consider, not concepts, but instances which are brought under universal concepts. So, they say that metal is wider than gold, since it contains more species than gold, and if we wish to enumerate the individuals made of gold on the one hand and those of metal on the other, the latter will be more than the former, which will therefore be contained in the latter as a part in the whole. By the use of this observation, and with suitable symbols, we could prove all the rules of logic by a calculus somewhat different from the present one -- that is, simply by a kind of inversion of it. However, I have preferred to consider universal concepts, i.e. ideas, and their combinations, as they do not depend upon the existence of individuals."

So, while Leibniz reasoned in relation to parts and wholes, he did so quite differently from the sense by which modern analytic philosophy declares mathematics to be extensional. In fact, this is the same distinction as found between Frege's concepts and his extensions of concepts (whence we should speak clearly of "comprehensionalist set theory" as opposed to intractible shorthands).

For what this is worth, the purely logical form for the principle of the identity of indiscernibles should probably be attributed to Aquinas. Leibniz generalized it. Although I have not searched for it, Leibniz indicates that Aquinas used the idea to explain how God could know each soul individually.

The justification for the principle lies with a section of Categories from Aristotle. The section discusses priority and simultaneity. Within it, Aristotle asserts that genera are prior to species. Aquinas took this to justify the idea that an individual is the lowest species.

Today, this resolves into the admonition by first-order advocates to not confuse an individual with a singleton. It finds expression in the debate between set theory and mereology. Russell had emphasized the import of Peano's recognition of the need to make the distinction. In the modern literature, Potter has reiterated Russell's position while accusing Dedekind of mereological conflation. Lawvere has elevated Dedekind while claiming that philosophy has wandered from mathematical principles.

Kant's distinction between logic and mathematics is expressed once again in his "Amphiboly of Concepts of Reflection." It is here where the distinction is grounded upon the principle of the identity of indiscernibles. At about A263/B319 Kant writes:

"Identity and Difference -- If an object is presented to us on several occasions but always with the same inner determinations, then it be taken as object of pure understanding, it is always one and the same, only one thing, not many. But, if it is appearance, we are not concerned to compare concepts, difference of spatial position at one and the same time is still an adequate ground for the numerical difference of the object."

Kant goes on to discuss Leibniz as he understood the problem (differing philosophies of Leibniz and Newton). He claims that Leibniz mistakenly took appearances (phenomena) as things-in-themselves (noumena). Kant maintains that plurality and numerical difference are ground in "the given" if sensible intuition.

His paragraph concludes:

"For one part of space, although completely similar and equal to another part, is still outside the other, and for this reason is a different part, whichh when added to it constitutes with it a greater space. The same must be true of all things which exist simultaneously in the different spatial positions, however similar and equal they may otherwise be."

The last passage quoted shows why it had been easy for Hilbert to compare his metamathematical enumeration by strokes with Kantian ideas. It may also be compared with Lawvere's "bag of dots" justified by Cantorian notions disputed by the logicists.

For what this is worth Dr. Katz, I had not been arguing with anyone. I thought I was answering the original poster's question.

@OP

Shortly after posting an answer (two parts) at the link,

https://philosophy.stackexchange.com/questions/91773/which-philosophy-topics-are-necessary-for-philosophy-of-mathematics

PSE instituted a logon policy. That original poster accepted my answer because of the advice it had contained. The stackexchange community does not like me very much.

You might consider the advice. Good luck in all of your endeavors.

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