Questions tagged [mathematics]

Use this tag for questions a mathematician would feel at home answering and can be traced back to an area of mathematics.

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11 votes
3 answers
303 views

Why are impredicative constructions used less in type theory than in material set theory?

Many infinitary objects in (say) ZFC are constructed with impredicative principles. The natural numbers are formed by intersecting every inductive set (whose existence is given by the axiom of ...
  • 3,133
20 votes
1 answer
268 views

Examples of formalisation of abelian categories

The question I would be interested to hear about examples of formalisation of the theory of abelian categories in theorem provers, and in particular formalisations of things like the zig-zag lemma and ...
5 votes
0 answers
1k views

What's the general idea (intuition) behind Andrej's topos where reals are countable? [closed]

I am very surprised that Andrej Bauer claimed to have found a topos where reals are countable. He said it'll take a month to write down the proof, and I think that's very understandable. However, I'm ...
  • 5,374
14 votes
4 answers
282 views

Algorithms obtained through constructive formalization

Formal proofs in proof systems that avoid the law of the excluded middle and certain other principles can be automatically converted into algorithms. What useful new algorithms have been produced by ...
16 votes
1 answer
516 views

Proof assistants and formalised mathematics in the MSC database

I was looking at the MSC2020 database and I find it hard to identify a field suitable for works about proof assistants and formalised mathematics. 03B70 ("Logic in Computer Science") might ...
22 votes
3 answers
346 views

To what extent is formalized mathematics publishable?

I am interested in contributing to the formalization of mathematics, but I don't know the extent to which such activities are publishable. Here are some questions in this vein: How can one determine ...
2 votes
2 answers
182 views

What is known about minimal sets of axioms? [closed]

There are several axioms that are known to be independent of the usual ones; for instance, the Axiom of Choice. This axiom can be stated in several equivalent ways, e.g.: For every set $A$, $\mathcal{...
  • 443
8 votes
1 answer
113 views

Have the common algorithms in existing computer-assisted proofs been transitioned to proof assistants?

In my experience, one of the most common applications of computer-assisted proofs in math is establishing that a given system of constraints is infeasible. Here are some examples of settings and tools ...
19 votes
1 answer
322 views

How hard is computing integrals in Lean?

Are there tools in mathlib which let you give computations of integrals which would roughly follow standard methods for solving them? For now let me restrict attention to some undergrad-level ...
  • 988
8 votes
1 answer
120 views

What are some useful resources for a mathematician interested in learning Isabelle/HOL?

What are some useful and reliable resources for a mathematician interested in learning Isabelle/HOL? Could be online (websites) or physical (books).
  • 1,258
13 votes
2 answers
148 views

verifying combinatorial constructions - choice of a proof assistant

The choice of the proof assistant to use for formalisation depends on the area quite a bit; e.g. they say that algebraic topology comes easy in HoTT assistants. What would be the most natural choices ...
11 votes
1 answer
138 views

Representing $\Bbb RP^2$ in Lean: building a type representing a particular set

I need to work with the set of all lines in the Cartesian plane. For my context, the natural way to think of this is that a line can be described by an equation $Ax + By + C = 0$, where $A$ and $B$ ...
  • 305
11 votes
2 answers
112 views

Proof assistants with dynamic scope/local instances/etc.?

Say I'm formalizing something in group theory, and I'm working with some action $\cdot$ of $(G, +)$ on a set. In my math textbook, the identity of $\cdot$ is explicitly mentioned once (if that), and ...
27 votes
8 answers
538 views

Where can I find lists of theorems that have been verified?

I recall many years ago seeing a very large and well-interlinked (by computer) list of verified results starting from base assumptions and leading to all sorts of things that naive me did not expect ...
26 votes
3 answers
294 views

Examples of new mathematics discovered through formalization?

In his essay Why formalize mathematics?, Patrick Massot discusses several reasons behind why a working mathematician might be interested in proof formalization. One of the the reasons he discusses is, ...
19 votes
0 answers
219 views

Can we automatically get around set-theoretic difficulties?

One of the main technical annoyances of working with (large) categories is the variety of set-theoretic difficulties that come about with it: if we use ZFC as background logic, then those large ...
  • 988
27 votes
2 answers
780 views

What set-theoretic definitions can't easily be formalized in a type theory?

Most proof assistants (with some exceptions like Isabelle/ZF or the B method) rely on type theory. See also the MathOverflow question What makes dependent type theory more suitable than set theory for ...
  • 417
43 votes
2 answers
1k views

Are some proof assistants better suited for given areas of math than others?

There are many different proof assistants out there, and I think it is reasonable to expect that more or less all results we prove in everyday mathematics can be proven in any of them. One nice way to ...
  • 988
23 votes
2 answers
525 views

Why haven't all of the "hundred greatest theorems" been formalized yet?

The "hundred greatest theorems" used to be maintained here. See this page for the current status of the formalization of these theorems. Apparentely, Fermat's Last Theorem, the Isoperimetric ...
17 votes
2 answers
201 views

How can I prove facts about floating point calculations?

In computing, there is a standard called IEEE 754. Unlike in mathematics, computers have to fit numbers into a finite amount of space; to do this, they use a format a bit like scientific notation, ...
  • 443
12 votes
1 answer
137 views

Exotic natural language summaries of formal proofs

Mathematical proofs written in natural language can often be used as guides to create formal proofs in proof assistants, depending on the level of detail of the proof and how many results and concepts ...