On David Hilbert’s “On the Infinite”
(“Über das Unendliche”)
Page last updated 27 Feb 2023
In a 1925 lecture the mathematician David Hilbert set out his philosophy of the infinite. (Footnote:
David Hilbert, Über das Unendliche, Mathematische Annalen 95.1, 1926, 161-190, translation by Erna Putnam and Gerald J. Massey, viewable online at Philosophy of mathematics: On the infinite by Daivd Hilbert.)
While a few quotations from this work are regularly rehashed out of context, the details of Hilbert’s overall viewpoint espoused therein are not widely known, and so
At the outset, Hilbert acknowledges that the meaning of the infinite in mathematics has still never been completely resolved:
“… in spite of the foundation Weierstrass has provided for the infinitesimal calculus, disputes about the foundations of analysis still go on. These disputes have not terminated because the meaning of the infinite, as that concept is used in mathematics, has never been completely clarified.”
He proceeds to set out what he considers to be the real nature of the infinite in mathematics. He first examines the question of whether the infinite occurs in nature, and comes to a conclusion which is precisely the opposite to that of Cantor, who asserted that the infinite, and even the transfinite, occurs in nature, see, for example Cantor’s belief in the reality of the infinite. Hilbert decides that the infinite does not occur at all in nature:
“… the infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to.”
“We have established that the universe is finite in two respects, i.e., as regards the infinitely small and the infinitely large.”
“Our principal result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought …”
Today it is generally accepted that Hilbert was right and Cantor was wrong; we have found no instances of an infinity of any kind in nature, never mind any “bigger” infinities, nor is there any indication that we might discover any sort of infinity in the future. But although Hilbert thought it worthwhile to clarify his ideas on this matter, he seems to have turned a blind eye to the possibility that Cantor’s beliefs in that respect involved fundamental difficulties that led to an inherently problematic theory of the infinite.
How to determine if a new concept is mathematically acceptable?
He then considers how a decision as to whether a new concept may be deemed acceptable as a new part of mathematics, and asserts that “success” is to be the final arbiter:
“If, apart from proving consistency, the question of the justification of a measure is to have any meaning, it can consist only in ascertaining whether the measure is accompanied by commensurate success. Such success is in fact essential, for in mathematics as elsewhere success is the supreme court to whose decisions everyone submits.”
However, this simply begs the question:
What is success in mathematics?
Is it simply a vote by mathematicians that it is successful, and hence a purely subjective appraisal? That is not acceptable; mathematics, of all subjects, must strive for as much objectivity as possible. Of course, the success of those parts of mathematics that are used in the advancement of science and technology can be subjected to a reasonably objective appraisal - but what are we to do with those parts of mathematicians that have never had any successful real world application? Are we to simply take the word of the mathematicians who study those parts of mathematics when they claim that their mathematics is successful? We wouldn’t accept that in any other sphere of human activity, so why should we do so for mathematical concepts that have never had any useful application? If politicians are approached for funding, in no other area would they unquestioningly accept the claims of those who ask for funds.
Praise for Cantor’s set theory: Potential and Actual Infinity
Hilbert then pauses to lavish praise upon Cantor’s set theory, and asserts that Cantor’s notion that there are two different types of infinity is correct:
“In this paper, we are interested only in that unique and original part of set theory which forms the central core of Cantor’s doctrine, viz., the theory of transfinite numbers. This theory is, I think, the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity. What, then, is this theory?
Someone who wished to characterize briefly the new conception of the infinite which Cantor introduced might say that in analysis we deal with the infinitely large and the infinitely small only as limiting concepts, as something becoming, happening, i.e., with the potential infinite. But this is not the true infinite. We meet the true infinite when we regard the totality of numbers 1, 2, 3, 4, … itself as a completed unity, or when we regard the points of an interval as a totality of things which exists all at once. This kind of infinity is known as actual infinity.”
This is arrant nonsense. To talk about the “infinitely large” as a limiting concept is a blatant contradiction, since to be infinitely large is to be nothing other than to be without any limitation to size. And to lump the “infinitely large” and the “infinitely small” together as though their properties are comparable is, if not intentionally dishonest, completely misleading, since the notion of “infinitely small” does imply a limit (zero), whereas the notion of “infinitely large” does not.
Furthermore, the very phrase potential infinite (“potentiellen Unendlichen”) is also nonsensical, being an obvious oxymoron. Either a thing is without any limit, or it is restricted by some sort of limit. To say that a thing is potentially without a limit could only mean one of two things:
- It is inherently without any limit, in which case, the inclusion of the term potential is superfluous and only serves to confuse, so the term should simply be “infinite”, or
- It requires something that changes its properties, so that the thing changes from being something that has a limit to something that does not have a limit, in which case when the term is applied to something, it means, ipso facto, that without such change the thing is not unlimited, and again the term only serves to confuse, and the term in this case should simply be “finite”.
The notion of actual infinity (“aktual Unendlichen”) can also be summarily dismissed as a similarly incoherent notion. Every number in any set of natural numbers is associated back to the natural number zero by a finite number of unitary iterations. If there could be more than a finite number of natural numbers actually together in a set, then there would have to be numbers that are so associated with the natural number zero by more than a finite number of unitary iterations. But that is impossible, since such an entity, by definition, cannot be a natural number.
These are simple straightforward refutations that should be readily accessible to anyone who expends a modicum of contemplative effort. But mathematicians like Hilbert instead refuse to see the obvious, like a lover blinded by love for the object of his desire.
There is also a more detailed look at the notion of these two types of infinity at Actual, Completed and Potential Infinity.
Limitlessly large sets of different sizes
Hilbert then goes on to talk about limitlessly large sets that are “larger” than other limitlessly large sets:
“But it was Cantor who systematically developed the concept of the actual infinite. Consider the two examples of the infinite already mentioned
- 1, 2, 3, 4, …
- The points of the interval 0 to 1 or, what comes to the same thing, the totality of real numbers between 0 and 1.
It is quite natural to treat these examples from the point of view of their size. But such a treatment reveals amazing results with which every mathematician today is familiar. For when we consider the set of all rational numbers, i.e., the fractions 1⁄2, 1⁄3, 2⁄3, 1⁄4, … 3⁄7, … , we notice that - from the sole standpoint of its size - this set is no larger than the set of integers. … Surprisingly enough, the set of all the points of a square or cube is no larger than the set of points of the interval 0 to 1. … On learning these facts for the first time, you might think that from the point of view of size there is only one unique infinite. No, indeed! … the set (2) cannot be enumerated, for it is larger than the set (1).”
Here Hilbert falls into the same fallacy as so many others, simply assuming that the absence of any one-to-one correspondence between two sets necessarily implies that one of the limitlessly large sets has fewer elements than the other limitlessly large set, that one set is larger than the other. This is not only an intuitive assumption that has absolutely no logical basis - it is actually self-contradictory. It is truly astonishing that so many mathematicians and logicians have a blind spot in this respect and refuse to see the obvious. For an in-depth analysis on this matter, see One-to-one Correspondences and Properties. (Footnote: Also see Proof of more Real numbers than Natural numbers? and Why do people believe weird things?)
Criticism of Cantor’s set theory
Having previously heaped praise on Cantor’s set theory, Hilbert proceeds to point out all the contradictions that are inherent in that theory, seemingly totally oblivious to the incongruity of his stance:
“In the joy of discovering new and important results, mathematicians paid too little attention to the validity of their deductive methods. For, simply as a result of employing definitions and deductive methods which had become customary, contradictions began gradually to appear. These contradictions, the so-called paradoxes of set theory, though at first scattered, became progressively more acute and more serious. In particular, a contradiction discovered by Zermelo and Russell had a downright catastrophic effect when it became known throughout the world of mathematics. … Too many different remedies for the paradoxes were offered, and the methods proposed to clarify them were too variegated. Admittedly, the present state of affairs where we run up against the paradoxes is intolerable. Just think, the definitions and deductive methods which everyone learns, teaches, and uses in mathematics, the paragon of truth and certitude, lead to absurdities! If mathematical thinking is defective, where are we to find truth and certitude?”
And after pointing out that mathematics has immersed itself into a quagmire of self-inflicted contradictions, he offers his view of how the problem might be resolved:
“There is, however, a completely satisfactory way of avoiding the paradoxes without betraying our science. The desires and attitudes which help us find this way and show us what direction to take are these:
- Wherever there is any hope of salvage, we will carefully investigate fruitful definitions and deductive methods. We will nurse them, strengthen them, and make them useful. No one shall drive us out of the paradise which Cantor has created for us.
- We must establish throughout mathematics the same certitude for our deductions as exists in ordinary elementary number theory, which no one doubts and where contradictions and paradoxes arise only through our own carelessness.”
This emotive proclamation “No one shall drive us out of the paradise …” sits very uneasily alongside his concomitant claim that every cause of any contradiction in Cantor’s set theory will be rooted out without compunction. It indicates very clearly a strong desire to retain aspects of that theory that are emotionally appealing. It is not far-fetched to suggest that this emotional attachment led to the turning of a blind eye to the possibility that the notion of a number larger than any limitlessly large number might be indicative of a fundamental problem worth investigating in depth.
From Hilbert’s 1920 Göttingen lectures, transcribed by Moses Schonfinkel and Paul Bernays, with annotations by Hilbert, translation by William Ewald, in the book “From Kant to Hilbert”, Vol 2, Oxford University Press, 1996.
In 1908 Poincaré said of set theory:
“Unfortunately they have reached contradictory results, what are called the Cantorian antimonies, to which we shall have occasion to return. These contradictions have not discouraged them and they have tried to modify their rules so as to make those disappear which had already shown themselves, without being sure, for all that, that new ones would not manifest themselves. It is time to administer justice on these exaggerations. I do not hope to convince them; for they have lived too long in this atmosphere. Besides, when one of their demonstrations has been refuted, we are sure to see it resurrected with insignificant alterations, and some of them have already risen several times from their ashes. Such long ago was the Lernaean hydra with its famous heads which always grew again.”
From introductory paragraphs added in 1908, to a 1905 article in the “Revue de metaphysique et de morale” under the title “Les mathematiques et la logique”, translation by William Ewald, in the book “From Kant to Hilbert”, Vol 2, 1996.)
Furthermore, the flagrant fact is that in spite of the so-called Cantorian “fruitful scientific” approach, over 100 years later there has been no scientific application of any of the notions of Cantorian set theory that cannot be achieved by a simple theory where elements are distinct from sets, and sets cannot be elements, see Natural Set Theory. (Footnote:
As Nik Weaver said:
“Virtually all modern mathematics outside set theory itself can be carried out in formal systems which are far weaker than Zermelo-Fraenkel set theory and which can be justified in very concrete terms without invoking any supernatural universe of sets … axiomatic set theory is not indispensable to mathematical practice, as most philosophers of mathematics have apparently assumed it to be. It is one arena in which mathematics can be formalized, but it is not the only one, nor even necessarily the best one.”
From: ‘The Concept Of A Set’, arXiv:0905.1677, 2009.) (Footnote: As Solomon Feferman said:
“I am convinced that the Platonism which underlies Cantorian set theory is utterly unsatisfactory as a philosophy of our subject … Platonism is the medieval metaphysics of mathematics; surely we can do better.”
From: ‘Infinity in Mathematics: Is Cantor Necessary?’ in “Infinity in Science”, Instituto dello Enciclopedia Italiana (1987), pp.151‑209, also in the book: In the Light of Logic, OUP on Demand, 1998.) Poincaré was remarkably prescient - today, over 100 years later, the contradictions of set theory have still not been eradicated; I document many of them on this website. The initial flurry of attempts of the early 20th century to remove all contradictions from set theory has settled down into the dismal defeatist dogma that it is impossible to remove these contradictions.
So, we can now ask, after almost 100 years since Hilbert wrote these words, has the sought-after certitude been achieved? Have all contradictions and paradoxes been expunged from set theory?
No - it is patently obvious that this is not the case.
Perhaps the reason is simply because, as Hilbert himself remarked, there is not a real understanding of the infinite:
“Obviously these goals can be attained only after we have fully elucidated the nature of the infinite.”
Isn’t it the case, that despite the constant forlorn hope that set theory has been on the right track for over 100 years, mathematicians have still not fathomed how to deal with set theory and the infinite without generating contraindications? For more on set theory, see the overview starting at Overview of set theory: Part 1: Different types of set theories.
Logic or intuition?
Towards the end of his article, Hilbert expounds his ideas of how to create a completely consistent formal mathematical system based entirely on logical symbol manipulation. But then, astonishingly, having explained his completely logical formal system, he turns about face and declares that logic is not sufficient and that intuition is always necessary:
“In contrast to the earlier efforts of Frege and Dedekind, we are convinced that certain intuitive concepts and insights are necessary conditions of scientific knowledge, and logic alone is not sufficient.”
But intuition cannot be anything more than a launching pad for ideas which must then withstand any assault of logic. Intuition is nothing more than a temporary ladder that must be thrown away once it has been replaced by an indestructible logical staircase.
It is clear that Hilbert failed to achieve his objective of clarifying the infinite as it is used in mathematics. Several fundamental errors of logic, as indicated above, together with an emotional attachment to various concepts, eliminated any possibility that he might attain a lucid objective assessment of the situation.
Having examined various works by Gödel, Hilbert, Cantor and others, it is evident that having an expertise in the manipulation of mathematical symbols bears no correlation to a capacity for profound philosophical thinking outside of that specialized area. Indeed the level of philosophical thinking outside of their area of expertise strikes one as having the sort of naivety that is engendered by a careful avoidance of any real engagement with the key points in case the result might not be to one’s liking - much as a child, when faced with accumulating evidence that Santa Claus might not be a real person, takes refuge in the comfort of anything positive, such as the morning absence of the food left out on Christmas Eve, and pushes any negative thoughts to the back of the mind, to be left unexamined.
Moreover, it is evident that any criticism of set theory that is found in various journals is indulgent of the claims of conventional mathematics. Instead of continuing to hammer away relentlessly at all the deficiencies of set theory, there seems to be an unwritten gentleman’s agreement that the obvious flaws will be overlooked in a pretense that any problems are minor rather than fundamental.