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I finally wrote this: intuitionistic mathematics.
Good, recently I was equipping somebody else’s writings here with hyperlinks and didn’t know which entry the term “intuitionistic mathematics” should point to. I ended up making it point to intuitionistic logic.
I have added the floating TOC to your entry now.
Great. I looked a little at intuitionism and constructivism more generally around 20 years ago. It’s good to catch up again.
Is it worth saying something on what is coalgebraic about intuitionistic maths? I see i quoted there Bauer on fans and spreads as terminal coalgebras.
Is it worth saying something on what is coalgebraic about intuitionistic maths?
My ideas on that are pretty vague, but at least I can link to that comment in the article.
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?
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?
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?
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?
Is that the intuitionist intuition? That only continuous constructions can cope with non-lawlike objects? Why?
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”
Has Per Martin-Löf tied together his work on algorithmic randomness and on constructive type theory?
The words “lawlike” and “predeterminate” don’t appear when searching either of those theses. Can you point to where they discuss intuitionistic mathematics?
I’m looking for an actual discussion of how such modalities relate to Brouwer’s intuitionistic mathematics.
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.
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.
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