Zeno's paradox (at least as it is usually presented) is not a formal mathematical problem — GrandMinnow
I watched the one about Zeno's paradox up to the point you said, "mathematicians begin with an assumption [that] [an] infinite process can be completed". What exact particular quote by a mathematician are you referring to? Please cite a quote and its context so that one may evaluate your representation of it in context, let alone your generalization about what "mathematicians [in general] begin with as an assumption" — GrandMinnow
Again, set theory rises to the challenge of providing a formal system by which there is an algorithm for determining whether a sequence of formulas is indeed a proof in the system. So whatever you think its flaws are, that would have to be in context of comparison with the flaws of another system that itself rises to that challenge. — GrandMinnow
It's worth noting that the challenges in the first post of this thread have been met. But hell if I know whether the poster understands that by now. — GrandMinnow
The mainstream real line is a vast darkness speckled by bright computable numbers, numbers we can actually talk about, numbers with names, while most of them are lost in the darkness and inferred to exist only indirectly. — norm
But when I do math, I don't think of R in terms of that glorious set-theory mess at all...In my POV, foundations is its own fascinating kind of math. It doesn't really hold up the edifice of applied calculus, IMO. It's a decorative foundation. Humans trust tools that work most of the time. Full stop. — norm
To me it is concerning that the foundations are so disconnected from the applications. Could this be an indication that further foundational work is required? — Ryan O'Connor
To me it is concerning that the foundations are so disconnected from the applications — Ryan O'Connor
Could this be an indication that further foundational work is required? — Ryan O'Connor
I'm no expert, but my impression was the math moved toward being totally mechanized, totally formal, totally computer-checkable — norm
It's unfortunate the word "foundations" is used in mathematics. Foundational set theory (that intersects with philosophy significantly) and all that revolves around it is a marvelous intellectual area, but a great deal of mathematics flows unimpeded by its pronouncements. Do most seasoned mathematicians worry about the existence of irrational or non-computable numbers? Or non-measurable sets? Not likely. — jgill
Afterwards the prof - a young energetic fellow - made the statement, "Unless you are really fascinated with the material in this course, I recommend you never take another foundations of mathematics class." — jgill
Do you believe that infinite processes cannot be completed? — Ryan O'Connor
Set theory itself (at least at the level of this discussion), as formal mathematics, does not say "an infinite process can be completed". Set theory doesn't even have vocabulary that mentions "completion of infinite processes". And the assumptions of set theory are the axioms. There is no axiom of set theory "an infinite process can be completed". — GrandMinnow
infinite sum — Ryan O'Connor
Imagine having a discussion with a child — Ryan O'Connor
If they ask a question, one way of addressing it is to add layers of complexity to the issue such that it is beyond their grasp — Ryan O'Connor
pile on a dozen textbooks — Ryan O'Connor
and say 'ask me when you know what you're talking about' — Ryan O'Connor
I'm here to learn — Ryan O'Connor
We don't need gatekeepers — Ryan O'Connor
we need people to help the litterers learn how not to litter — Ryan O'Connor
feel free to ignore my messages — Ryan O'Connor
if you're inclined to help then I welcome it — Ryan O'Connor
I don't think you will enjoy us talking informally about potential [problems with the current philosophical foundations for math] — Ryan O'Connor
If you don't want to talk informally, or if you want to disregard Zeno's paradox due to its informal presentation that is fine — Ryan O'Connor
If set theory properly lies at the foundation of mathematics then (I believe) it should have no loose threads (e.g. paradoxes). — Ryan O'Connor
And I'm sure Norm would agree, that movement would drive most mathematicians out of the profession. It can't be emphasized enough how much mathematics depends on intuition, imagination, inventiveness, and a spirit of exploration. Devising and proving theorems is an art form. — jgill
You might be interested in this perspective as it offers a different perspective (granted, probably wrong and certainly half-baked). Nevertheless, I'd love to hear your thoughts on this view, especially if you can find flaws in it...but no pressure at all! — Ryan O'Connor
the foundations — Ryan O'Connor
set theory — Ryan O'Connor
LoL. I'm sure the philosophy students in the room were aghast. — Ryan O'Connor
You don't know anything about it.
Yet you have persistent critiques of it.
How do you do it? — GrandMinnow
You might be interested in this perspective as it offers a different perspective (granted, probably wrong and certainly half-baked). Nevertheless, I'd love to hear your thoughts on this view, especially if you can find flaws in it...but no pressure at all! — Ryan O'Connor
With a certain aplomb. I admire his spirit while avoiding his critiques. :cool: — jgill
At what point did we prove that it was a number? — Ryan O'Connor
That would be the empty set, Ryan. We were all math majors and the course was taught in the math department. — jgill
That's a well made video. FWIW, I do like the continua-based approach. Have you looked into smooth infinitesimal analysis ? It seems similar. One issue worth noting is your description of a quasi-Riemann integral as an endless process. In an actual Riemann integral, for f which is continuous on [a,b], there exists a definite sum. In other words, we know that it's a particular real number, even if we only ever approximate it (like the areas under the standard normal curve.)
In SIA, certain issues are circumvented, because every function is smooth (infinitely differentiable). Some strange logic is involved. — norm
Interesting! So, you think the square root of 2 could be something other than a number. Well, the square root operation is closed over real numbers i.e. a square root of a real number has to be a real number. Does that answer your question? — TheMadFool
I have not looked into SIA — Ryan O'Connor
However, just a few points on the wiki page seem concerning to me, like I have no problems with discontinuous functions but I do have a problem with infinitesimals. — Ryan O'Connor
For Riemann integrals, how do we know that it corresponds to a real number if we are only ever able to approximate it? — Ryan O'Connor
Weird stuff, IMHO. Low priority in the world of mathematics. — jgill
I do have a problem with infinitesimals — Ryan O'Connor
What argument have I lost? — Metaphysician Undercover
"Existence" is a word which is being used here as a predicate. — Metaphysician Undercover
So we need criteria to decide which referents have existence in order justify any proposed predication. Naturally we ought to turn to the field of study which considers the nature of existence, to derive this criteria, and this is metaphysics. — Metaphysician Undercover
Mathematics does not study the nature of existence, so mathematicians have no authority in this decision as to whether something exists or not, — Metaphysician Undercover
regardless of whether it is a common opinion in the society of mathematicians. — Metaphysician Undercover
If you are arguing otherwise, then show me where mathematics provides criteria for "existence" rather than starting with an axiom which stipulates existence. — Metaphysician Undercover
Sabine Hossenfelder has a current blog post on Do Complex Numbers Exist? Might be relevant, I'm not qualified to judge. — Wayfarer
ah OK - a hit, and a miss! — Wayfarer
But metaphysicians don't know any more about existence than the rest of us. — fishfry
Metaphysics studies questions related to what it is for something to exist and what types of existence there are. Metaphysics seeks to answer, in an abstract and fully general manner, the questions:[3]
What is there?
What is it like?
Topics of metaphysical investigation include existence, objects and their properties, space and time, cause and effect, and possibility. Metaphysics is considered one of the four main branches of philosophy, along with epistemology, logic, and ethics.[4] — Wikipedia: Metaphysics
The entire history of mathematics is filled with examples, starting from the discovery of irrational numbers right through to the present day. — fishfry
hat's been the case in my experience too. For applications, though, the dual numbers are actually important today. Some of the autodifferentiation powering machine learning use the forward method, employing dual numbers to great effect (and even hyperdual numbers.) This allows one to compute f(x) and grad(f(x)) at the same time at low cost. — norm
but felt that she entirely missed the meaning of complex numbers — fishfry
My newest guess is, that there is a layer for all objects (quants) that can interact
(except interacting with gravity),
and if some interact than the layer for all objects is increased.
This way we get in the layers a kind of time arrow since the big bang,
and properties (even in math like prime decomposition)
can depend from it and change with time. — Trestone
"Existence" is a word which is being used here as a predicate. — Metaphysician Undercover
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