added pointer to:

- Arend Heyting,
*Intuitionism: An introduction*, Studies in Logic and the Foundations of Mathematics, North-Holland (1956, 1971) [ISBN:978-0720422399]

added the missing list of original articles on formalizing intuitionism (all grabbed and polished-up from *BHK interpretation*, where their relevance is dubious):

Arend Heyting,

*Die formalen Regeln der intuitionistischen Logik. I, II, III.*Sitzungsberichte der Preußischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse (1930) 42-56, 57-71, 158-169abridged reprint in:

Karel Berka, Lothar Kreiser (eds.),

*Logik-Texte*, De Gruyter (1986) 188-192 [doi:10.1515/9783112645826]Arend Heyting,

*Die intuitionistische Grundlegung der Mathematik*, Erkenntnis**2**(1931) 106-115 [jsotr:20011630, pdf]Andrey Kolmogorov,

*Zur Deutung der intuitionistischen Logik*, Math. Z.**35**(1932) 58-65 [doi:10.1007/BF01186549]Hans Freudenthal,

*Zur intuitionistischen Deutung logischer Formeln*, Comp. Math.**4**(1937) 112-116 [numdam:CM_1937__4__112_0]Arend Heyting,

*Bemerkungen zu dem Aufsatz von Herrn Freudenthal “Zur intuitionistischen Deutung logischer Formeln”*, Comp. Math.**4**(1937) 117-118 [doi:CM_1937__4__117_0]L. E. J. Brouwer,

*Points and Spaces*, Canadian Journal of Mathematics**6**(1954) 1-17 [doi:10.4153/CJM-1954-001-9]Georg Kreisel, Section 2 of:

*Mathematical Logic*, in T. Saaty et al. (ed.),*Lectures on Modern Mathematics III*, Wiley New York (1965) 95-195

Changing a link name to now point article for a topic that was previously previously (before 2020) non-existent.

Brouwer-Kripke scheme => Kripke’s schema

It seems both ways of writing it (BK resp just K) are still in use.

Related note: I think there’s many “schema”’s in math, but authors (mostly non-English authors I conjecture) like to write “scheme” for such “schema”’s (not to be confused with affine scheme type of schemes, etc.) I’m not 100% sure when the latter is appropriate (I’m Viennese myself), but I think “schema” is generally the right way to to in those situations.

]]>With regard to #5, Frank was referring to #14 in this thread I think.

]]>There is also this relevant thread, but I’ll not merge this one in as it’s not exclusively to do with the page intuitionistic mathematics.

]]>I have merged together three threads which all were to do with the page intuitionistic mathematics. The comments 5-12 comprised one of these threads.

The comments 1-4, 13-28 comprised the second of the threads. In fact comment 13 was originally made 45 minutes earlier than is now recorded on the server (at 03:59:16 server time), and actually came before comments 11 and 12, but I have moved it 45-ish minutes later (to 04:46:00) to make the merge cleaner.

The comment 29 was the only entry in the third thread, created earlier today.

]]>added pointer to:

- Hermann Weyl,
*Über die neue Grundlagenkrise der Mathematik*Zürich 1920 (purl:PPN266833020_0010)

The "different line of research" is NuPrl-like? It would be useful if you added references. Perhaps even to intuitionistic mathematics or constructive mathematics. ]]>

It may be unnecessarily confusing to try to make the connection between sheaf-models-of-choice-sequences and sheaf-toposes-for-realizability in the sense of Awodey and Bauer. Sheaf models of choice sequences are about taking sheaves over either some space or some category of well-behaved spaces, and using the resulting toposes to study the relationship between lawlike and non-lawlike operations and objects; this is the tradition of Fourman (“continuous truth”), Troelstra, many others, etc. In these sheaf toposes, the non-constructive Brouwerian principles (like continuity, bar induction, fan theorem, etc.) hold.

The paper of Awodey and Bauer on the other hand uses sheaf toposes in a different way (i.e. not over a notion of space whose open sets give finitary data about choice sequences), as a means to study general realizability: you render a PCA into a site, and obtain a Kripke-Joyal forcing semantics for realizability. One of the results is that you can embed the modest sets for the PCA into this grothendieck topos.

Now, in a very different line of research, others have studied a totally different connection between realizability and Brouwerian/continuous truth, which is where you basically “do realizability” over a space or some category of spaces. A purpose of doing this might be to connect Brouwer’s notion of truth, based on infinitary well-founded trees trees (and did not *ever* involve the notion of an algorithm in the sense formalized by a PCA, in apparent contradiction with the so-called BHK interpretation) with the BHK interpretation of intuitionistic truth.

The relation between lawlike and computable is made for instance in Kreisel and Troelstra - elimination of choice sequences. This work has later been connected to continuous truth. The relation between realizability and sheaf models has been made by Awodey and Bauer. I am not sure all the dots have been connected though. I don’t know of any work directly relating modal logic and lawlike sequences.

]]>I’m looking for an actual discussion of how such modalities relate to Brouwer’s intuitionistic mathematics.

]]>The words “lawlike” and “predeterminate” don’t appear when searching either of those theses. Can you point to where they discuss intuitionistic mathematics?

]]>He has often linked the type theory and measure theory, but I don't know whether he ever made a formal connection. ]]>

Has Per Martin-Löf tied together his work on algorithmic randomness and on constructive type theory?

]]>From the latter

the modal logic for local maps in the case of RT(A,A#) and RT(A#) can be seen as a modal logic for computability

where RT(A,A#) is a realizability topos

]]>which one intuitively can think of as having “continous objects and computable morphisms”

This was later connected to the logic of local toposes by Awodey and Birkedal; see modal type theory. ]]>

Is *that* the intuitionist intuition? That only continuous constructions can cope with non-lawlike objects? Why?

Thanks.

I just happened to be reading Brower’s cambridge lectures, and it occurred to me that he seems to use qualifiers like “lawlike” and “predeterminate” like a (comonadic) modality. Which makes sense when thinking about how in spatial/cohesive type theory the function with domain $\flat A$ don’t have to be “continuous”, and similarly how in intuitionistic mathematics it’s the presence of the non-lawlike objects (free choice sequences) that’s supposed to force things to be continuous. Has anyone tried to formalize Brouwerian intuitionistic mathematics using a modality like this?

]]>I moved some references from intuitionistic mathematics to constructive mathematics because as far as I could see they were not at all about Brouwerian intuitionistic mathematics but rather about constructive mathematics more generally (though sometimes using the label “intuitionistic”).

Is the Kleeny-Vesley topos really about *intuitionistic mathematics* in Brouwer’s sense?

I think I was wondering if there was something more ’philosophical’ to say. I see you joined in that discussion at the Café. I guess I was wondering if anything coalgebraic was to be seen in Brouwer’s thinking.

How about the ’Second act of Intuitionism’?

Brouwer’s second act of intuitionism gives rise to choice sequences, that provide certain infinite sets with properties that are unacceptable from a classical point of view. A choice sequence is an infinite sequence of numbers (or finite objects) created by the free will. The sequence could be determined by a law or algorithm, such as the sequence consisting of only zeros, or of the prime numbers in increasing order, in which case we speak of a lawlike sequence, or it could not be subject to any law, in which case it is called lawless. Lawless sequences could for example be created by the repeated throw of a coin, or by asking the creating subject to choose the successive numbers of the sequence one by one, allowing it to choose any number to its liking. Thus a lawless sequence is ever unfinished, and the only available information about it at any stage in time is the initial segment of the sequence created thus far. Clearly, by the very nature of lawlessness we can never decide whether its values will coincide with a sequence that is lawlike. Also, the free will is able to create sequences that start out as lawlike, but for which at a certain point the law might be lifted and the process of free choice takes over to generate the succeeding numbers, or vice versa. (SEP: Intuitionism)

Are these not streams?

]]>(NB added in 2021 following the thread merger mentioned in #30: This comment originally followed #4.) Just came across this thread while looking for something else. The relation to coalgebras and streams processors is well-understood. eating trees. Does that fit with what you have in mind?

]]>Thanks. Where you have the external link to an article it would be better to instead list this article in the list of reference at the bottom of the entry, equip it with an anchor, and point to that. That’s more informative (the reader may see the author and title of the article without having to follow a link) and more robust (the external link might break).

]]>