Every now and then I have seen the following idea, but I don't know the original author. I am interested in order to give credit to him.

Every object of thought that can be identified, i.e., distinguished from all other objects of thought, exists in at least one memory or processing unit, i.e., in the spatio-temporal physical universe. Therefore it has a lot of rational spatio-temporal coordinates exclusively of its own. Take one of them to enumerate the object. Since the set of rational spatio-temporal coordinates, even in an infinite and eternal universe, is countable and since the set of thought objects can be injected into that countable set, the set of all thought objects is countable.

My question: Who was the first to recognize this fact?

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    $\begingroup$ I do not see how something as vague as "object of thought" can be identified, let alone supplied with coordinates and/or injected into a countable set, but you can look at Lullian Circle, "a paper machine operated by rotating concentrically arranged circles to combine his symbolic alphabet, which was repeated on each level. These combinations were said to show all possible truth about the subject of inquiry". $\endgroup$ – Conifold Apr 25 '17 at 17:56
  • $\begingroup$ @Conifold: I do not share your scepsis. You are right that "object of thought" may be something vague. Of interest however is only that every object of thought that qualifies for an appearance in mathematics (like thoughts that later may be written as symbol, number, set, axiom, theorem, or spoken or painted) is not vague. Since the set of all objects of thought injects into the countable set of rational coordinate quadruples the subset of sufficiently defined objects of thought is countable with certainty. $\endgroup$ – user5693 Apr 25 '17 at 20:22
  • $\begingroup$ This is a discussion for Philosophy SE, but here is Bertrand Russell:"If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." $\endgroup$ – Conifold Apr 25 '17 at 21:44
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    $\begingroup$ I'm pretty sure the idea that only countably many descriptions (or thoughts, or whatever) can be described (or exhibited, or whatever) in any reasonably unique way was "in the air" during 1895-1905, and probably various versions of this idea can be found in the writings of Borel, Lebesgue, Peirce, Hadamard, Russell, etc. from this time --- look through Moore's Zermelo's Axiom of Choice book and the references here, among other places. $\endgroup$ – Dave L Renfro Apr 26 '17 at 14:25

While it is true that from the modern viewpoint something as vague as an "object of thought" can hardly be identified, as correctly pointed out in the comments, such creatures were used by no less an authority than Richard Dedekind to "prove" the existence of an infinite set. The idea is essentially a recursive construction involving thinking about the previous thought. This was discussed in detail here.

  • $\begingroup$ Thank you, but Dedekind (arguing essentially like Bolzano) did not assume a set of all thought objects. "There are infinite systems. Proof (a similar reflection can be found in § 13 of the Paradoxien des Unendlichen by Bolzano (Leipzig 1851)) . The world of my thoughts, i.e., the collection S of all things which can be objects of my thinking, is infinite. For, if s is an element of S, then the thought s' that s can be an object of my thinking is itself an object of my thinking." [R. Dedekind: "Was sind und was sollen die Zahlen?", 8th ed., Vieweg, Braunschweig (1960) p. 14] $\endgroup$ – user5693 Apr 26 '17 at 9:09
  • $\begingroup$ Dedekind's approach has been adapted by Zermelo: "But in order to save the existence of 'infinite' sets we need yet the following axiom, the contents of which is essentially due to Mr. R. Dedekind. Axiom VII." [E. Zermelo: "Untersuchungen über die Grundlagen der Mengenlehre I", Math. Ann. 65 (1908) p. 266f] Therefore we cannot tidentify this with the idea asked for in my question. $\endgroup$ – user5693 Apr 26 '17 at 9:14
  • $\begingroup$ This is correct, and it shows that Zermelo recognized that a new axiom is needed here, and also discarded Dedekind's reasoning. But what is interesting is that Dedekind does envision an infinite collection of thoughts. This is a modest step in the direction you propose in your question which is much more ambitious. $\endgroup$ – Mikhail Katz Apr 26 '17 at 9:22
  • $\begingroup$ If we are giving historical credit for infinity of thoughts shouldn't we go back to Leibniz's (Lullius inspired) characteristica universalis and the alphabet of human thought, which expresses it all and reduces thinking to calculation? $\endgroup$ – Conifold Apr 27 '17 at 0:30
  • $\begingroup$ @Conifold, very interesting. I was somewhat familiar with the characteristica universalis but I was not familiar with this aspect of Leibniz's work on thoughts. One thought is that you can try to reduce thinking to calculation without envisioning an infinite totality of thoughts, as Dedekind did. Did Leibniz envision such totalities? My impression was that, on the contrary, he thought that infinite totalities were contradictory (unlike infinite numbers, by the way, and contrary to the syncategoremania that we sought to refute here. $\endgroup$ – Mikhail Katz Apr 27 '17 at 9:08

I believe the desired answer is Jules Richard and

the Richard antinomy (all things which form objects of thought are defined through finitely many words, therefore denumerable; on the other hand the set of all real numbers is already nondenumerable)

The above is a translation by W. Boos (1995, p. 285) of the Jahrbuch review of Hermann Weyl’s Habilitationsvortrag (1910). It is an oversimplified paraphrase of both Weyl and Richard’s original paradox (1905), which was technically precise and about “numbers”, not “things”. Also, like @DaveLRenfro it replaces the OP’s “rational spatio-temporal coordinates” by “verbal definitions”. Last twist, Boos attributes the quoted review to Skolem, but the online Jahrbuch says Salkowski!

  • $\begingroup$ I don't think that this answers the question which asks for "this enumeration" meaning the rational spatio-temporal coordinates. Further, objects of thought do not only include numbers, but also things that can only insufficiently be expressed in words like mental pictures, dreams or feelings of fear or happiness or certain kinds of satisfaction and many more. Meanwhile I have found a nice definition of thought-object. Since it is too long for this comment I have added it to my answer. $\endgroup$ – Otto Jun 27 '17 at 11:29
  • $\begingroup$ That's why I quoted Richard’s paradox in this 1910 “folk” version. It literally says objets of thought, and of course enumerating with words includes the option of enumerating with rational coordinates, since those are expressible through finitely many words. It also predates Mückenheim by decades. $\endgroup$ – Francois Ziegler Jun 27 '17 at 11:46
  • $\begingroup$ Sorry, I could not find "Gedankendinge" in the quotes of Weyl. I have no access to the translation. $\endgroup$ – Otto Jun 27 '17 at 11:49
  • $\begingroup$ Weyl (1910) has “Dinge, von denen man reden will”; the review (1910) has “Dinge, die Gegenstand des Denkens bilden”. $\endgroup$ – Francois Ziegler Jun 27 '17 at 11:53
  • $\begingroup$ By the way: Enumerating objects with words or numbers allows diagonalization to arrive at another object whereas enumerating all objects of the infinite and eternal universe excludes diagonalization since all "diagonal objects" (results of diagonalization) are enumerated already. $\endgroup$ – Otto Jun 27 '17 at 11:54

I have seen the argument of surjecting a countable set of spatio-temporal coordinates on the set of all ideas including all real numbers that ever have been or will be realized: Countability of the real numbers, p. 275 of https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf. Since no reference to earlier work is given, I assume this is the first appearance of the idea unless earlier authors can be identified.

EDIT(1): In the writings of Borel, Lebesgue, Peirce, Russell, or Moore's "Zermelo's Axiom of Choice" I have not been able to find this idea. Hadamard was anyway in favour of Zermelo's approach to set theory.

Edit(2): Here is a nice definition of thought-object:

At request of the referee who asked what is a thought-object let me add: I understand it to be a thought about an object which may exist or not. Thus it is an electrochemical event in the brain or/and its record in the memory. In particular it is a physical thing in space time. Of course it is difficult to characterise any physical phenomena. But we have the ability to recognize thoughts as identical or different, just as we have the ability to recognize a silent lightning from a thunderous one. Hence I understand Hilbert's words as follows: mathematicians imagine sets which do not exist, but their thoughts about sets do exist and they can arise prior to the thoughts of most elements in those sets. (J. Mycielski: "Russell's paradox and Hilbert's (much forgotten) view of set theory" in G. Link (ed.): "One hundred years of Russell's paradox: mathematics, logic, philosophy", de Gruyter, Berlin (2004) p. 534)


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