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We can't define "foundations of mathematics" without defining mathematics. My favorite stab at this is from Thurston's On Proof and Progress in Mathematics. Whi

by btilly 3y ago

We can't define "foundations of mathematics" without defining mathematics. My favorite stab at this is from Thurston's On Proof and Progress in Mathematics. Which may be read in full at https://www.math.toronto.edu/mccann/199/thurston.pdf https://www.math.toronto.edu/mccann/199/thurston.pdf: Could the difficulty in giving a good direct definition of mathematics be an essential one, indicating that mathematics has an essential recursive quality? Along these lines we might say that mathematics is the smallest subject satisfying the following: - Mathematics includes the natural numbers and plane and solid geometry. - Mathematics is that which mathematicians study. - Mathematicians are those humans who advance human understanding of mathematics. In other words, as mathematics advances, we incorporate it into our thinking. As our thinking becomes more sophisticated, we generate new mathematical concepts and new mathematical structures: the subject matter of mathematics changes to reflect how we think. The foundations of mathematics then refers to the essential core that we have based our mathematical thinking on. Which is something that most of us only wish to think about when our thinking has run into problems and we need a better foundation.

vlovich123 3y ago
Is mathematics just the study of all possible languages that are constructed from strict internally consistent rules?
- saalweachter 3y ago
  Mathematics is the study of things you can convince people are true without physical evidence.
  - adastra22 3y ago
    I like this one. Succinct and to the point. However this would also include religion :\
    - saalweachter 3y ago
      Two different religionists can have a conversation and agree on what is true?
      - vlovich123 3y ago
        Within the lens of the other religion probably. Except from hardcore zealots, religious people often agree on true things that they find in common between religions (eg idol worship being a sin, murder, adultery, etc etc).
    - btilly 3y ago
      Not if you added "...using only pure logic". Religions do not convince with pure logic.
      - munch117 3y ago
        The problem is that now you need to define what "pure logic" is. Here's my definition: Math is the study of intersubjective knowledge.
- btilly 3y ago
  No, because some possible languages are far more interesting than others.
  - bmacho 3y ago
    I don't think that's a problem: if there are uninteresting languages, then you must study them, then work hard to prove that they are uninteresting first. That is, mathematics indeed does study uninteresting languages. .. mathematics is the same for me as GP said: you have a set of symbols, a set of rules, now derive true statements. (Or at least this is a good view, but other views can be also good.)
    - btilly 3y ago
      Most mathematicians don't feel that way. There are uninteresting languages that can be considered, but nobody considers them unless a reason arises to do so. Attempting to prove that they are uninteresting is not itself an interesting problem, because it is easy to prove that such proofs do not generally exist. Indeed things become interesting not because they are inherently interesting in and of themselves. But because something else that is already interesting turns out to connect to the not yet interesting thing, and studying the not yet interesting therefore becomes relevant to the interesting one.
      - vlovich123 3y ago
        The uninteresting languages are still part of mathematics though. It’s just that we choose not to explore them because we don’t think there’s any value. If you limit it just to the languages that we find interesting, then “what is mathematics” is a fluid definition depending on what we choose to focus on today whereas my definition is a bit more durable. Also, on the topic of “uninteresting”, if I recall correctly number theory (or discrete math) was explored by a mathematician claiming it’ll never have any application and that’s why he was doing it, only for cryptography to come by within a century after that. At least if I recall the apocryphal story told to me in class correctly. Erdos seems like he’d line up time wise and focus wise but I can’t find any quotes from him to that effect so maybe I’m misrembering?
- adastra22 3y ago
  So Japanese grammar is math?
  - vlovich123 3y ago
    I think so, but it wouldn’t be very interesting. The rules of Japanese grammar mathematically could only tell you if something is in the Japanese language or not. The other challenge is that human languages tend to be filled with exceptions and intentional violations of rules for creative effect. That kind of stuff doesn’t really fly in mathematical languages.
  - btilly 3y ago
    English grammar can be. Buffalo is an animal, a city, and a verb meaning "to bewilder and confuse". As a result the following can be proven. The word buffalo, repeated any number of times, can always be parsed into a valid English sentence. If the number of times it appears is prime, it can only be parsed into ONE possible English sentence. This is clearly a statement of mathematics!
    - adastra22 3y ago
      No, it is a mathematical statement about linguistics. It should not be surprising that mathematics can be used to describe things. Everything in the universe can be described by mathematics--even truly random, acausal events have probability distributions which are described by mathematics. But being described by mathematics does not make it mathematics. That's a type error.
Someone 3y ago
> Mathematics includes the natural numbers and plane and solid geometry. That, to me, feels similar to Euclid’s fifth postulate, in the sense that it isn’t something you’d want to include in a definition. Maybe, the similarity doesn’t end there, and we have multiple mathematics, just as pure mathematicians don’t talk about algebra, but about an algebra, with some of them being more interesting than others, and it not always being easy to show that two algebras are different from each other. For example, we have mathematics without the law of the excluded middle and with it, ones without the axiom of choice and with it, etc.
- andrybak 3y ago
  > For example, we have mathematics without the law of the excluded middle and with it, ones without the axiom of choice and with it, etc. This is a bit off-topic. Elsewhere in comments [1] people talk about Terence Tao and his recent work in Lean. Lean is based on intuitionistic logic, a base which doesn't include axiom of choice and the excluded middle. Since Tao and other classical mathematicians need excluded middle for proofs by contradiction. The Classical library in Lean starts by declaring axiom of choice [2] and after some lemmas introduces excluded middle as a theorem, called "em" for brevity [3]. I like how Lean makes this particular choice for classical mathematics very obvious. [1] https://news.ycombinator.com/item?id=38102096#38104412 https://news.ycombinator.com/item?id=38102096#38104412 [2] https://github.com/leanprover-community/lean/blob/cce7990ea86a78bdb383e38ed7f9b5ba93c60ce0/library/init/classical.lean#L13 https://github.com/leanprover-community/lean/blob/cce7990ea8... [3] https://github.com/leanprover-community/lean/blob/cce7990ea86a78bdb383e38ed7f9b5ba93c60ce0/library/init/classical.lean#L66 https://github.com/leanprover-community/lean/blob/cce7990ea8...
naasking 3y ago
Mathematics is the study of structure.
pilgrim0 3y ago
Correct me if I’m hallucinating, anybody. Doesn’t a foundation have to change and adapt in relation to what is grounded in it? If the foundation is to be all encompassing, to abstract enough to allow and absolutely account for every possible superstructure that it may ever ground, would not the foundation itself be contentless, inexpressive? “Silence contains all sound”, and “Every moment has an infinite amount of time” are constructs that came to my mind as isomorphic to the questioning. As mathematical structures gets finer and more intricate, purely on the basis of necessity (survival of the mathematician, mankind and mathematics itself) and experimentation, so their foundations must undergo at least some change? Even though such changes are purely virtual, and of interpretative nature? The same ideology, but “taken as” something else, thus, ultimately, subjective? There also seems to emerge an issue of ordering, of a demand for primality and derivation, which screams paradox. Could all of mathematics fit within itself, being itself employed to bootstrap its own, with no second system? A sort of ontological auto-genesis? Could our conception of paradox be itself primal, and perhaps, in some plane, could it be something ranking higher, of first-class? Also, I’ve been thinking, recently, on the role of time in structures. There can’t possibly be any structure whatsoever without time, or, more concretely, at least the memory of events, recollecting distinctive and contrasting entropic signatures. So, mathematics manifesting as, of, and for structure, wouldn’t it require, first and foremost, a treatment from physics? Regular or meta?
- bvssvni 3y ago
  In mathematics, the roof holds up the building, not the foundation. Since humans use mathematics a lot, we design foundations to our specific needs. It is not the building we are worried about, we just want better foundations to create better tools. Not only are we going to treat mathematics as subjective, but also having formal theories that reason about different notions of subjectivity. https://crates.io/crates/joker_calculus https://crates.io/crates/joker_calculus > Could our conception of paradox be itself primal, and perhaps, in some plane, could it be something ranking higher, of first-class? Yes! Paradoxes are statements of the form `false^a` in exponential propositions. https://crates.io/crates/hooo https://crates.io/crates/hooo > Also, I’ve been thinking, recently, on the role of time in structures. There can’t possibly be any structure whatsoever without time, or, more concretely, at least the memory of events, recollecting distinctive and contrasting entropic signatures. So, mathematics manifesting as, of, and for structure, wouldn’t it require, first and foremost, a treatment from physics? Regular or meta? Path semantical quality models this relation, where you have different "moments" in time which each are spaces for normal logical reasoning. Between these moments, there are ways to propagate quality, which is a partial equivalence. https://github.com/advancedresearch/path_semantics https://github.com/advancedresearch/path_semantics
yMEyUyNE1 3y ago
A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas” (Hardy, 1992 , p. 84) I lean towards patterns of ideas, relations, structures and organizations. edit: from https://maa.org/press/periodicals/convergence/mathematics-as-the-science-of-patterns-mathematics-as-the-science-of-patterns https://maa.org/press/periodicals/convergence/mathematics-as...

by btilly 3y ago

vlovich123 3y ago

Is mathematics just the study of all possible languages that are constructed from strict internally consistent rules?

saalweachter 3y ago
Mathematics is the study of things you can convince people are true without physical evidence.
- adastra22 3y ago
  I like this one. Succinct and to the point. However this would also include religion :\
  - saalweachter 3y ago
    Two different religionists can have a conversation and agree on what is true?
    - vlovich123 3y ago
      Within the lens of the other religion probably. Except from hardcore zealots, religious people often agree on true things that they find in common between religions (eg idol worship being a sin, murder, adultery, etc etc).
  - btilly 3y ago
    Not if you added "...using only pure logic". Religions do not convince with pure logic.
    - munch117 3y ago
      The problem is that now you need to define what "pure logic" is. Here's my definition: Math is the study of intersubjective knowledge.
btilly 3y ago
No, because some possible languages are far more interesting than others.
- bmacho 3y ago
  I don't think that's a problem: if there are uninteresting languages, then you must study them, then work hard to prove that they are uninteresting first. That is, mathematics indeed does study uninteresting languages. .. mathematics is the same for me as GP said: you have a set of symbols, a set of rules, now derive true statements. (Or at least this is a good view, but other views can be also good.)
  - btilly 3y ago
    Most mathematicians don't feel that way. There are uninteresting languages that can be considered, but nobody considers them unless a reason arises to do so. Attempting to prove that they are uninteresting is not itself an interesting problem, because it is easy to prove that such proofs do not generally exist. Indeed things become interesting not because they are inherently interesting in and of themselves. But because something else that is already interesting turns out to connect to the not yet interesting thing, and studying the not yet interesting therefore becomes relevant to the interesting one.
    - vlovich123 3y ago
      The uninteresting languages are still part of mathematics though. It’s just that we choose not to explore them because we don’t think there’s any value. If you limit it just to the languages that we find interesting, then “what is mathematics” is a fluid definition depending on what we choose to focus on today whereas my definition is a bit more durable. Also, on the topic of “uninteresting”, if I recall correctly number theory (or discrete math) was explored by a mathematician claiming it’ll never have any application and that’s why he was doing it, only for cryptography to come by within a century after that. At least if I recall the apocryphal story told to me in class correctly. Erdos seems like he’d line up time wise and focus wise but I can’t find any quotes from him to that effect so maybe I’m misrembering?
adastra22 3y ago
So Japanese grammar is math?
- vlovich123 3y ago
  I think so, but it wouldn’t be very interesting. The rules of Japanese grammar mathematically could only tell you if something is in the Japanese language or not. The other challenge is that human languages tend to be filled with exceptions and intentional violations of rules for creative effect. That kind of stuff doesn’t really fly in mathematical languages.
- btilly 3y ago
  English grammar can be. Buffalo is an animal, a city, and a verb meaning "to bewilder and confuse". As a result the following can be proven. The word buffalo, repeated any number of times, can always be parsed into a valid English sentence. If the number of times it appears is prime, it can only be parsed into ONE possible English sentence. This is clearly a statement of mathematics!
  - adastra22 3y ago
    No, it is a mathematical statement about linguistics. It should not be surprising that mathematics can be used to describe things. Everything in the universe can be described by mathematics--even truly random, acausal events have probability distributions which are described by mathematics. But being described by mathematics does not make it mathematics. That's a type error.

Someone 3y ago

> Mathematics includes the natural numbers and plane and solid geometry. That, to me, feels similar to Euclid’s fifth postulate, in the sense that it isn’t something you’d want to include in a definition. Maybe, the similarity doesn’t end there, and we have multiple mathematics, just as pure mathematicians don’t talk about algebra, but about an algebra, with some of them being more interesting than others, and it not always being easy to show that two algebras are different from each other. For example, we have mathematics without the law of the excluded middle and with it, ones without the axiom of choice and with it, etc.

andrybak 3y ago
> For example, we have mathematics without the law of the excluded middle and with it, ones without the axiom of choice and with it, etc. This is a bit off-topic. Elsewhere in comments [1] people talk about Terence Tao and his recent work in Lean. Lean is based on intuitionistic logic, a base which doesn't include axiom of choice and the excluded middle. Since Tao and other classical mathematicians need excluded middle for proofs by contradiction. The Classical library in Lean starts by declaring axiom of choice [2] and after some lemmas introduces excluded middle as a theorem, called "em" for brevity [3]. I like how Lean makes this particular choice for classical mathematics very obvious. [1] https://news.ycombinator.com/item?id=38102096#38104412 https://news.ycombinator.com/item?id=38102096#38104412 [2] https://github.com/leanprover-community/lean/blob/cce7990ea86a78bdb383e38ed7f9b5ba93c60ce0/library/init/classical.lean#L13 https://github.com/leanprover-community/lean/blob/cce7990ea8... [3] https://github.com/leanprover-community/lean/blob/cce7990ea86a78bdb383e38ed7f9b5ba93c60ce0/library/init/classical.lean#L66 https://github.com/leanprover-community/lean/blob/cce7990ea8...

naasking 3y ago

Mathematics is the study of structure.

pilgrim0 3y ago

Correct me if I’m hallucinating, anybody. Doesn’t a foundation have to change and adapt in relation to what is grounded in it? If the foundation is to be all encompassing, to abstract enough to allow and absolutely account for every possible superstructure that it may ever ground, would not the foundation itself be contentless, inexpressive? “Silence contains all sound”, and “Every moment has an infinite amount of time” are constructs that came to my mind as isomorphic to the questioning. As mathematical structures gets finer and more intricate, purely on the basis of necessity (survival of the mathematician, mankind and mathematics itself) and experimentation, so their foundations must undergo at least some change? Even though such changes are purely virtual, and of interpretative nature? The same ideology, but “taken as” something else, thus, ultimately, subjective? There also seems to emerge an issue of ordering, of a demand for primality and derivation, which screams paradox. Could all of mathematics fit within itself, being itself employed to bootstrap its own, with no second system? A sort of ontological auto-genesis? Could our conception of paradox be itself primal, and perhaps, in some plane, could it be something ranking higher, of first-class? Also, I’ve been thinking, recently, on the role of time in structures. There can’t possibly be any structure whatsoever without time, or, more concretely, at least the memory of events, recollecting distinctive and contrasting entropic signatures. So, mathematics manifesting as, of, and for structure, wouldn’t it require, first and foremost, a treatment from physics? Regular or meta?

bvssvni 3y ago
In mathematics, the roof holds up the building, not the foundation. Since humans use mathematics a lot, we design foundations to our specific needs. It is not the building we are worried about, we just want better foundations to create better tools. Not only are we going to treat mathematics as subjective, but also having formal theories that reason about different notions of subjectivity. https://crates.io/crates/joker_calculus https://crates.io/crates/joker_calculus > Could our conception of paradox be itself primal, and perhaps, in some plane, could it be something ranking higher, of first-class? Yes! Paradoxes are statements of the form `false^a` in exponential propositions. https://crates.io/crates/hooo https://crates.io/crates/hooo > Also, I’ve been thinking, recently, on the role of time in structures. There can’t possibly be any structure whatsoever without time, or, more concretely, at least the memory of events, recollecting distinctive and contrasting entropic signatures. So, mathematics manifesting as, of, and for structure, wouldn’t it require, first and foremost, a treatment from physics? Regular or meta? Path semantical quality models this relation, where you have different "moments" in time which each are spaces for normal logical reasoning. Between these moments, there are ways to propagate quality, which is a partial equivalence. https://github.com/advancedresearch/path_semantics https://github.com/advancedresearch/path_semantics

yMEyUyNE1 3y ago

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas” (Hardy, 1992 , p. 84) I lean towards patterns of ideas, relations, structures and organizations. edit: from https://maa.org/press/periodicals/convergence/mathematics-as-the-science-of-patterns-mathematics-as-the-science-of-patterns https://maa.org/press/periodicals/convergence/mathematics-as...