Axiomatics: Mathematical thought and high modernism
old.maa.org
old.maa.org
My best example of the split is https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives Wikpedia notes that "The list of unsuccessful proposed proofs started with Euler's, published in 1740,[3] although already in 1721 Bernoulli had implicitly assumed the result with no formal justification." The split between pure (Euler) and applied(Bernoulli) is already there.
The result is hard to prove because it isn't actually true. A simple proof will apply to a counter example, so cannot be correct. A correct proof will have to use the additional hypotheses needed to block the counter examples, so cannot be simple.
Since the human life span is 70 years, I face an urgent dilemma. Do I master the technique needed to understand the proof (fun) or do I crack on and build things (satisfaction)? Pure mathematicians are planning on constructing long and intricate chains of reasoning; a small error can get amplified into a error that matters. From a contradiction one can prove anything. Applied mathematics gets applied to engineering; build a prototype and discover problems with tolerances, material impurities, and annoying edge cases in the mathematical analysis. A error will likely show up in the prototype. Pure? Applied? It is really about the ticking of the clock.
[1] https://www.reddit.com/r/Bitcoin/comments/1pv8ty/selfish_min...
A type is a theorem and its implementation a proof, if you believe that Curry-Howard stuff.
We “prove” (implement) advanced “theorems” (types) using already “proven” (implemented) bodies of work rather than return to “axioms” (machine code).
The example you gave concerns differentiation. Differentiation is messy in real analysis because it's messy in numerical computing. How real analysis fixes this mess parallels how numerical computing must fix the mess. How do we make differentiation - or just derivatives, perhaps - computable?
The rock-bottom condition for computability is continuity. All discontinuous functions are uncomputable. It turns out that it is sufficient, to make your theorem hold, to have the 2nd partial derivatives f_{xy} and f_{yx} be continuous. They wouldn't even be computable otherwise!
One of the proofs provided uses integration. In numerical contexts, it is integration which is considered "easy", and "differentiation" which is considered hard. This is totally backwards to symbolic calculus.
The article also mentions Distribution Theory. This is important in the theory of linear PDEs. I suspect it is implicit in the algorithmic theory as well, whether practitioners have spelled this out or not. This is a theory that makes the differentiation operator itself computable, but at the cost of making the derivatives weaker than ordinary functions. How so? On the one hand, it allows to obtain things like the Dirac delta as derivatives, but those aren't even functions. On the other hand, these objects behave like functions - let's say f(x,y) - but we can't evaluate them at points; instead, we can take their inner product with test functions, which we can use to approximate evaluation. This is important because PDE solvers may only be able to provide solutions in the weak, distribution-theoretic sense.
Functions define distributions, but not all distributions are defined that way, like the Dirac delta or integration over a subset.
This is incorrect. In mathematics there is a single definition of function. There is no conflict or contradiction. In all cases a function is a subset of the cross product of two spaces that satisfies a certain condition.
What changes from subject to subject is what the underlying spaces of interest are.
I'm not sure I understand what you mean here. I need some clarification. How does this have any bearing on whether functionals count as functions or not? What is the "underlying spaces of interest" in this example?
In some trivial way, every mathematical object can be seen as a function. You can replace sets in axiomatic set theory with functions.
Some books will say: a functional is a linear map….
Note that a linear map is a function.
This situation is a consequence of how mathematicians haven't always been sure how to define certain concepts. See "generating function" for yet another usage of the word "function" that's in direct contradiction with the last two. So that's three incompatible usages of the term "function". All this terminology goes back to the 1700s when mathematics was done without the rigour it has today.
I find it aggravating how you're so confidently wrong. I hope it's not on purpose.
[edit] [edit 2: Removed insults]
This is a fine example of irony.
Let V be a vector space over the reals and L a functional. Let v be a particular element of V. L(v) is a real number. It is a single value. L(v) can't be 1.2 and also 3.4. Thus L is a function.
A function is simply a subset of the product of two sets with the property that if (a,b) and (a, c) are in this subset then b=c.
Can you find a functional that does not meet this criterion? If so then you have an object such that L maps v to a and also maps v to c with a and c being different elements.
Find me a linear map that does not meet the definition of function. Give an example of a functional in which the functional takes a given input to more than one element of the target set.
I think you are not a mathematician and you also don't appear to understand that a word can have different meanings based on context. "generating function" isn't the same thing as "function". Notice that generating is paired with function in the first phrase.
Example: Jellyfish is not a jelly and not a fish. Biologists have got it all wrong!
> I think you are not a mathematician
Guess again.
> Example: Jellyfish is not a jelly and not a fish. Biologists have got it all wrong!
You have a problem with reading comprehension. I never said any mathematician was wrong.
Think about namespaces for a moment, like in programming. There are two namespaces here: The analysis namespace and the foundations namespace.
In either of those two namespaces, the word "mapping" means what you're describing: an arbitrary subset F of A×B for which every element of a ∈ A occurs as the first component in a unique element (x,y) ∈ F.
But the term "function" has a different meaning in each of the two namespaces.
The word "function" in the analysis namespace defines it to ONLY EVER be a mapping S -> R or S -> C, where S is a subset of C^n or R^n. The word "function" is not allowed to be used - within this namespace - to denote anything else.
The word "function" in the foundations namespace defines it to be any mapping whatsoever.
Hopefully, now you'll get it.
Interesting. So you think there are functions in real analysis that are studied that don't meet the definition I gave? Is there a functional that does not meet the definition I gave?
In all contexts a function is a subset of the product of two sets that meets a certain condition. Anything that does not meet this definition is not called a function.
Every functional meets the definition of function.
In real analyis one is interested in functions from R^n to R. They don't define function to be only something from R^n to R. It's just that these are the functions they wish to study. They don't define function to exclusively be a map from R^n to R. It’s just that these are the types of functions they care about.
No mathematician can possibly think function is anything other than a subset of the product of two spaces that meets a certain condition.
I see that you are desperately trying to distinguish "foundational" and "analysis" contexts from each other. If you are writing a book about analysis, it might be helpful to clarify that in this context you reserve "function" for mappings into ℂ or ℝ, for example [1] defines "function" exclusively as a mapping from a set S to ℝ (without any further requirements on S such as being a subset of ℝⁿ). Note that even under this restricted definition of function, a distribution still is a function.
In a general mathematical context, "function" and "mapping" are usually used synonymously. It is just not the case that such use is restricted to "foundations" only.
It seems to me that squabbles about issues like this are becoming more frequent here on HN, and I am wondering why that is. One hypothesis I have is that there is an influx of people here who learn mathematics through the lens of programs and type theory, and that limits their exposure to "normal" mathematics.
[1] Undergraduate Analysis, Second Edition, by Serge Lang
> I see that you are desperately trying to distinguish "foundational" and "analysis" contexts from each other
They literally are different. The proof is all the people here saying that distributions aren't functions, while displaying a clear understanding of what a distribution is. Maybe no one's "wrong" as such, if they're defining the same word differently.
I think you're the naive one here. Terminology is used inconsistently, and I tried to simplify the dividing line between different uses of it. I agree it's inaccurate to say it's decided primarily by Foundations vs Analysis, but I'm not sure how else to slice the pie. It's like how the same word can mean slightly different things in French and English. I agree it's quibbling, but it's harder to teach maths to people if these False Friends exist but don't get pointed out.
I never expected some obsessive user to make 6 different replies to one of my comments. Wow. This whole thing thread was a bit silly, and someone's probably going to laugh at it. I need to take another break from this site.
> I agree it's inaccurate to say it's decided primarily by Foundations vs Analysis, but I'm not sure how else to slice the pie.
Seems you agree with me after all.
> I agree it's quibbling, but it's harder to teach maths to people if these False Friends exist but don't get pointed out.
A distribution is a function, but considered on a different space.
It is even harder to teach math to people by insisting that above fact is wrong. Schwartz got a Fields medal for this insight.
Terry Tao writes in his analysis book:
Functions are also referred to as maps or transformations, depending on the context.
Tao certainly knows more about this than I ever will.
You have 6 posts in the thread started by my top comment. I had multiple replies to one of your posts because HN requires one to wait a while to reply and I was in a hurry. The order of posts doesn’t matter. At least not to me.
Insinuating I’m obsessive has a negative connotation. Along with outright insults such comments make you look bad and unreasonable.
Functions are also referred to as maps or transformations, de- pending on the context.
This after defining a function in essentially the same I did.
[My bad, it was Matvei, not Manuel, no idea how i mixed that up..
Checkout his childrens books, as well as
Note how the independent diagonals are what i consider interesting]
[I'm currently pondering how the "main diagonal" of a transition matrix provides objects, while all the off-diagonal elements are the arrows. This implies that by rotating into an eigenframe (diagonalising), we're reducing the diversion to -∞ (generalised eigenvectors have nothing to lose but their Jordan chains) and hence back in the world of classical boolean logic?]
L: quantal (quasiparticles)
Lagniappe: https://www.sciencedirect.com/science/article/pii/0022404993...
EDIT: I'm afraid I'm just learning fns vs distributions (curried fns?) myself.
I wonder how quasiparticles might relate to ideals (nuclei in quantale-speak I believe)? Note that something very much like quasiparticles is how regexen turn exponential searches into polynomial...
I ought to get overly emotional (in a bittersweet way) about all this, and i almost did, but Teddy reminded me to stay ataraxic (i.e. keeping his role in formulating key management policies purely in the cortex )
thank you for that blogpost about MPB (its one small step for fuzzablekind!)
[as well as the nuclei hint, more tk]