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Fuzzyness in maths

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Groupoid:
In response to a post in the SSSS Scriptorium.
Delicious discussion. I like it.


--- Quote from: JoB on February 16, 2021, 05:57:53 PM ---Gödel's work pinpoints that a) there is such an undecidable statement in every moderately advanced mathematical model, hence b) an infinite number of (truly different) models to choose from, and thus c) need for an infinite number of experiments (read: cannot be done) to pinpoint the proper model for an application case. If that doesn't make the relation between reality and math(ematical modeling) "fuzzy", I don't know what would.

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Boldness by me. I haven’t thought of the incompleteness theorem creating new models to go with the unprovable statements. I’m not well versed in the details, but the proof on Wikipedia seems to give non-constructive existence proofs of these models. Which is a bit sad.
Anyway. I meant "doing math" and/or "the internal stuff going on in math" is not so fuzzy. Of course the applicability of maths to the real world is a whole mess. How do we even know that abstract logical sentences have real-world applications? etc. I’m not so interested in the philosophy of that.
With the final argumentation I have to agree. But strangely I consider these examples (by feeling) relatively clear in the sense, that we know and can tell what’s going on. Another example from algebra: "5=0" is undecidable in the theory of fields/rings.

I’m sorry for introducing the term "fuzzy". If I had to define it now, I’d say "something is fuzzy" iff "the thing is a grey area" as mentioned by thegreyarea... But that doesn’t help much. I feel a little embarrassed to note, that I was (and still am) writing & thinking much quicker than I can check what I write. But that’s probably natural for philosophy.

I think either all of the above examples have to be called "fuzzy" or all have to be called "non fuzzy", since your argument extends fuzzy-ness.
For the act of doing math (proving a statement, giving an example, calculating, ...), I think none of these examples is "fuzzy"/unclear if one is able to get the definitions straight etc.
The fuzzy-ness comes in, when we have to choose what objects we try to model. i.e. do we only want to study ZFC-models, or also ZF-models? (Similar for geometry, algebra, ...) If one isn’t sure what (logical) models are (informal) models of something real, then this is the fuzzy-ness you claimed (I think). Especially in the sentence with "infinite experiments".
And with set-theory the choice of axioms is especially difficult, because in practice we very rarely concern us with the intricacies of large infinite sets. There are probably few experiments with physical stuff we can do to decide the existence of large cardinals etc.
(That seems stupid: what physical experiment could decide the continuum hypothesis for "our universe"? Nope, not thinking about it.)

Conclusion (maybe): The definition of grey areas in maths is itself a grey area.

Hair-splitting remark:

--- Quote from: JoB on February 16, 2021, 05:57:53 PM ---I'm fairly sure that the proponents of the latter still wouldn't have considered "just pick one already!" as the revelation of hidden truths that they were hoping for ... ::)

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Well... as the examples of the different geometries or of rings vs. fields show, sometimes it’s useful not to pick either. This is probably exactly what abstraction is about. To give general statements that hold for as many (logical) models as possible/feasible/useful/necessary/adequate.

And Argh! The editor has no undo-button. I lost a few lines and this made everything longer.

JoB:

--- Quote from: thegreyarea on February 16, 2021, 07:05:39 PM ---I hope it's all in the past.

--- End quote ---
(I think it is, but I'm not sufficiently in the math research loop anymore to confirm. When you're sorta hoping for a divine "truth behind truth", with statements that have a property of "being true" that math and logic just cannot get nailed down, the (mathematical) proof that you can forever extend your initial model by accepting either a Gödel statement or its negation as a new axiom is a quite final blow.)


--- Quote from: Groupoid on February 16, 2021, 07:24:16 PM ---I think a big problem with these flaming philosophical discussions about logic is, that while discussing we think we’re very sure we know what we mean, but actually we don’t express what we mean exactly. Or we don’t understand exactly what the other person meant. [...] No wonder mathematicians developed such a precise language to communicate their thoughts.

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Story from my university days, which saw the dpt. of mathematics getting its own new building after being housed in the offices and lecture halls of other dpt.s: The non-mathematicians overseeing the construction of the new building had the idea to save some pennies by installing smaller blackboards into the lecture halls, "because mathematical formulae are much more compact than the sentences of natural(*) language other departments have to write down". The math profs replied that, for sake of precision, they're required to write everything down, rather than just jotting down some key points like most other profs did. The dispute was settled when the janitor was asked to have a look at the blackboards, and found those which had been perused for math lectures for a couple years to be significantly more worn down than others. ;D

(*) Fightin' words for every true mathematician, that term is. ;)


--- Quote from: Groupoid on February 16, 2021, 07:15:28 PM ---I’m not well versed in the details, but the proof on Wikipedia seems to give non-constructive existence proofs of these models. Which is a bit sad.

--- End quote ---
Well, I don't think that a constructive proof is anywhere near the horizon for that kind of question; at least not in the sense that you can construct a model's Gödel statement. If you could, you'd have an algorithm to churn out a binary tree of models and their Gödel statements rooted in your initial model mechanistically, and we'd be working on a math of entire trees of models, instead of investigating a single model or a series (indexed by natural numbers) of models ...


--- Quote from: Groupoid on February 16, 2021, 07:15:28 PM ---Anyway. I meant "doing math" and/or "the internal stuff going on in math" is not so fuzzy.

--- End quote ---
... it usually isn't. As much as mathematics is trying to get its terms nailed down, the philosophers can still raise some brain-wrecking questions about the concepts it works with ...


--- Quote from: Groupoid on February 16, 2021, 07:15:28 PM ---Of course the applicability of maths to the real world is a whole mess. How do we even know that abstract logical sentences have real-world applications? etc. I’m not so interested in the philosophy of that.

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Agreed. In these days, we leave most of the number crunching to computers and their implementation of floating point numbers. Due to the finite, often fixed, number of bits used to represent those numbers, they not only have a limited range of values, but also varying precision; i.e., things happen like (simplified) computing "one million plus one" and getting "one million" for a result. In other words, computers' floating point numbers are not rings, groups, a mod-n arithmetic, etc. etc., and thus not subject to any classical number theory in the first place. A computation that actually isn't math, and not in the sense of "there's no actual value of \infty involved" ...


--- Quote from: Groupoid on February 16, 2021, 07:15:28 PM ---Another example from algebra: "5=0" is undecidable in the theory of fields/rings.

--- End quote ---
Of course, and mod-5 arithmetics (whether with integers or reals) is the specific and relevant case where it holds true.


--- Quote from: Groupoid on February 16, 2021, 07:15:28 PM ---I think either all of the above examples have to be called "fuzzy" or all have to be called "non fuzzy", since your argument extends fuzzy-ness.
For the act of doing math (proving a statement, giving an example, calculating, ...), I think none of these examples is "fuzzy"/unclear if one is able to get the definitions straight etc.
The fuzzy-ness comes in, when we have to choose what objects we try to model. i.e. do we only want to study ZFC-models, or also ZF-models? (Similar for geometry, algebra, ...) If one isn’t sure what (logical) models are (informal) models of something real, then this is the fuzzy-ness you claimed (I think). Especially in the sentence with "infinite experiments".

--- End quote ---
Ummmmhh that's what I pointed out as a specific example, yes. Nonetheless, the fact that models can be extended endlessly and in contradictory ways IMHO did make mathematics "fuzzier" in a sense. Before Gödel - see the discussion of "unreachable truths/falsehoods" -, mathematicians had a tendency to imagine that their models would change incessantly, but in the sense of evolution, yielding ever-"better" theories that will replace the older ones except for fringe cases. Gödel sunk that concept of "nearer, my God, to thee", if I may be so blasphemic. :)


--- Quote from: Groupoid on February 16, 2021, 07:15:28 PM ---And with set-theory the choice of axioms is especially difficult, because in practice we very rarely concern us with the intricacies of large infinite sets. There are probably few experiments with physical stuff we can do to decide the existence of large cardinals etc.
(That seems stupid: what physical experiment could decide the continuum hypothesis for "our universe"? Nope, not thinking about it.)

--- End quote ---
Who is "we" here? "The science of infinity" is one rather widely accepted definition of mathematics, even though it suggests that "computing" (with actual numbers) fails to qualify as "math".

But yes, our currently "established" understanding of the "real" (physical) world does not allow for actual infinite (finite universe wedged between Big Bang and Big Crunch) or infinitesimal (Heisenberg Uncertainty Principle) aspects. You have to go "(infinite) multiverse" (or "truth behind the truth" in a "God hides behind quantum uncertainty" sense) or somesuch to get reality up to "infinities only, please"-style math's muster. >:D


--- Quote from: Groupoid on February 16, 2021, 07:15:28 PM ---And Argh! The editor has no undo-button. I lost a few lines and this made everything longer.

--- End quote ---
The forum/website's builtin editor may not have one, but there's an Edit -> Undo menu point in my browser (Firefox) that seems to work satisfyingly as such ...

Groupoid:

--- Quote from: JoB on February 17, 2021, 04:22:20 AM ---The dispute was settled when the janitor was asked to have a look at the blackboards, and found those which had been perused for math lectures for a couple years to be significantly more worn down than others. ;D

--- End quote ---
:))


--- Quote from: JoB on February 17, 2021, 04:22:20 AM ---Agreed. In these days, we leave most of the number crunching to computers and their implementation of floating point numbers. Due to the finite, often fixed, number of bits used to represent those numbers, they not only have a limited range of values, but also varying precision; i.e., things happen like (simplified) computing "one million plus one" and getting "one million" for a result. In other words, computers' floating point numbers are not rings, groups, a mod-n arithmetic, etc. etc., and thus not subject to any classical number theory in the first place. A computation that actually isn't math, and not in the sense of "there's no actual value of \infty involved" ...

--- End quote ---
Hrm, the example fits well, but I wouldn’t call floating point arithmetic "not math" just because the most basic assumptions of algebra & arithmetic are not satisfied. (e.g. associativity) The example I thought about is the general assumption of most engineers that we live in R^3. At first sight I find it weird, that it is such a great approximation. Sometimes I’m even astonished by the fact, that the abstract "You have 10 sheep and I give you 5 more, then you have 15 sheep afterwards" actually applies to reality.


--- Quote from: JoB on February 17, 2021, 04:22:20 AM ---Of course, and mod-5 arithmetics (whether with integers or reals) is the specific and relevant case where it holds true.

--- End quote ---
And soo~oo many others where it is false. ;)


--- Quote from: JoB on February 17, 2021, 04:22:20 AM ---Ummmmhh that's what I pointed out as a specific example, yes. Nonetheless, the fact that models can be extended endlessly and in contradictory ways IMHO did make mathematics "fuzzier" in a sense. Before Gödel - see the discussion of "unreachable truths/falsehoods" -, mathematicians had a tendency to imagine that their models would change incessantly, but in the sense of evolution, yielding ever-"better" theories that will replace the older ones except for fringe cases. Gödel sunk that concept of "nearer, my God, to thee", if I may be so blasphemic. :)

--- End quote ---
Ah, I forgot the historical context. Yes, I think the whole "relativism" I’m used to (i.e. all theorems are "just" implications) must have developed greatly from Gödel’s work. Hilbert’s program...


--- Quote from: JoB on February 17, 2021, 04:22:20 AM ---Who is "we" here? "The science of infinity" is one rather widely accepted definition of mathematics, even though it suggests that "computing" (with actual numbers) fails to qualify as "math".

--- End quote ---
The use of "we" looks very wrong to me now. I meant non-set-theorists, or when-not-doing-set-theory, based on some essay I read. But I’ll have to find it again. (Appeal to authority. nice :() I think an example was "all Hilbert-spaces of cardinality P(P(P(N))) lack some property" or "some property doesn’t hold for all Hilbert-spaces of cardinality greater-or-equal to P(P(P(N)))", but I’m not sure. I got the example wrong. There’s a correction in a later post.
It wasn’t just about there being physically infinite things, but the author posed that most mathematics could be formalized "predicatively", without having an unconstrained power-set axiom [1]. And this somehow made them conclude "most mathematics (that doesn’t directly involve set-theory) can be done with cardinalities smaller than some cardinality".
I’ll look it up.


--- Quote from: JoB on February 17, 2021, 04:22:20 AM ---But yes, our currently "established" understanding of the "real" (physical) world does not allow for actual infinite (finite universe wedged between Big Bang and Big Crunch) or infinitesimal (Heisenberg Uncertainty Principle) aspects. You have to go "(infinite) multiverse" (or "truth behind the truth" in a "God hides behind quantum uncertainty" sense) or somesuch to get reality up to "infinities only, please"-style math's muster. >:D

--- End quote ---
I haven’t thought of Heisenberg as a lower bound on our "approach to the infinitesimal by measuring". That’s a nice thought I think.
I have a hard time digesting the second sentence. When I think I grasp your meaning, I agree. The many quoted things (thanks for quoting) are too undefined for me, so I’m not sure whether I understand what you mean.


--- Quote from: JoB on February 17, 2021, 04:22:20 AM ---The forum/website's builtin editor may not have one, but there's an Edit -> Undo menu point in my browser (Firefox) that seems to work satisfyingly as such ...

--- End quote ---
That’s a helpful tip, thanks. It’s always nice to know a little more about the tools/software I use. Browsers (I also use Firefox) have so many small & useful tools that are easily overlooked.

P.S.
[1] : https://math.stanford.edu/~feferman/papers/predicativity.pdf by Feferman says on p. 12 “In axiomatic Zermelo-Fraenkel set-theory, the fundamental source of impredicativity is the Separation Axiom scheme, ...” and goes on to describe how the axiom of infinity is also important. So as usual with axiom systems, there is not always a single logical culprit, but at least one that we might "feel" to be the reason.
I don’t think that this was the essay I talked about, but I can’t find something that fits better. The example with the hilbert spaces of some size is not in it... So maybe I mixed some papers up.

However that about the essay/paper might be. I believe that a lot of mathematics can be "done" (i.e. a lot of proofs can be carried out) with very weak set-theoretic/logical assumptions or concerns about the continuum hypothesis. This belief is founded (at least) in the above paper’s reference/discussion of Weyl, which I haven’t read myself. And I incorporated it implicitly & rather clumsily in the sentence with the "we".

moredhel:

--- Quote from: Groupoid on February 17, 2021, 04:02:18 PM ---Hrm, the example fits well, but I wouldn’t call floating point arithmetic "not math" just because the most basic assumptions of algebra & arithmetic are not satisfied. (e.g. associativity)

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It ist math, because it ist made of exactly defined operations on a set of numbers. But it does not work as most people would expect. E. g. the result of this little code:
float number = (float)1.0;
number = number/(float)3.0;
number = number + (float)1.0;
number = number - (float)1.0;
number = number * (float)3.0;
is that number equals 1.0000001.
Whenever rational numbers can be used to do wahtever you want, you should avoid floats.

Groupoid:
JoB I found it! “Is set theory indispensable” by Nik Weaver. On ArXiv and on his uni-webpage.

moredhel: That’s what I had in mind. Luckily we can choose to use rationals and reals in maths. But sometimes the intricacies of number-representations in computers can’t be avoided in practice. For the seemingly boring usual reasons of performance and memory use.

Edit: Fixed a typo.

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