You could narrow the discussion to whether college philosophy departments have a future. Where I taught that department was some time ago folded into a History, Geography, Philosophy, and Political Science Department. No undergrad degree in philosophy, but a minor is available.

If asked the same question about mathematicians, I'm not sure how I would answer. It's tempting to cite academia and graduate degrees, but there may be exceptions. Probably being published in PR journals, but some of my teaching colleagues avoided this, even as doctorates. Not clear.

Are you sure David?
I would think that what is "clear and easily comprehended" is uninteresting and boring, — A Realist

What you would think is wrong: Fermat's Last Theorem

However, the world of mathematics is so immense and in part has become so abstract that one mathematician might not be able to convey to another mathematician a recent result without some difficulty.

You write very well, but for those of us who have limited capacities for reflecting and processing it might help to break apart and separate very lengthy paragraphs, and/or do a bit more summarizing or condensing. You do have interesting insights. Did you say you worked in the health sector? An MD? Nurse? Just curious.

They can and should, we just have to fond some alternate means to their happiness — Pfhorrest

Is the "we" a branch of government? Or any other coercive agent?

So he's not an expert, just a self publicist. — Bartricks

You're probably correct. And if he reads this thread there is no danger he will show up.

All very illuminating. But what of the OP? :chin:

So when 2 is the member of a set, that is what the symbol "2" represents, an object, the number 2, which is independent of any group of two — Metaphysician Undercover

2={0,{0}} , 3={2,0,{0}}={0,{0},{0,{0}}} , etc. from the Peano Axioms through set theory. But there are other ways. Just a passing comment.

while there is obviously is a correct solution: because fixed point theorem proves that there exists a correct solution — ssu

I've worked with fixed points for a long time, mostly recreational now, but a few years ago one of my theorems was applied to decision making, specifically the psychology of groups involved. It surprised me. But I don't recall the title or authors, otherwise I would link it here for you to see. :cool:

I just received a spiritual message from Henry Fonda. He suggests you move on to Kirk Douglas, as Kirk feels neglected. :smile:

Should they be shielded from imbecilic comments and appalling reasoning? Why? — Bartricks

:lol:

I think that the incompleteness results have an effect on a wide range of things not just in the set theoretic realm and with the foundations of mathematics. We just don't want to make or are ignorant about the link to the incompleteness results.

I think the classic example of something being true but unprovable is a game theoretic situation where it's easy to show that a correct solution exists, yet there seems to be no way to get there. The existence of a correct solution can be shown...based on mathematics — ssu

As Nash demonstrated fixed-point theory is useful in game theory. Brouwer's fixed-point theorem was proven indirectly, with no simple path to its value, and this distressed Brouwer, who later turned to intuitionism. Proving a math object exists indirectly, but without a process for its construction, is still proving a theorem. This sort of thing has a superficial relation to Godel's works, but I don't think it's what he had in mind. Others here, with more knowledge of the matter can correct me if I'm in error.

↪jgill
How is he a 'professional'? — Bartricks

He's referred to as a British philosopher several times, and it appears he makes his living in his area of transhumanism. He seems to deal with ethical issues. But you guys can disown him if you wish. Makes no difference to me.

Mathematicians may be (but often aren't) predisposed to undertaking philosophical discussions - but solely in a fashion, that is adherent to quantitative frameworks. — Aryamoy Mitra

Topological ideas don't necessarily have a quantitative context. The concept of what continuous means in continuous transformations has been discussed at length in this forum.

For instance, Functions pertain to continuous variables across domains, — Aryamoy Mitra

Not necessarily. A basic definition lies in set theory and may be discrete.

It has the respect of most in the math community, but most of those think they will never come up against that roadblock. — jgill

In my view Gödel's incompleteness theorems, as the other incompleteness results, aren't roadblocks. — ssu

Set theory and foundation people certainly appreciate your perspective. And there are numerous examples of statements - conjectures - that can't be proven within, say, the Peano system, etc. Examples. Some probably are true.

But the fact remains that math people not in those areas are usually not very concerned, even if they are stumped in proving something. However, I haven't been around mathematicians for a long time and I could be mistaken.

That's an affirmative stance, but is it necessarily wise to accord a benefit of doubt, prior to witnessing an even partial demonstration of an argument's veracity? — Aryamoy Mitra

The benefit of doubt in the context of babble is inconsequential.

Virtual particles are described as temporary excitations of underlying quantum fields that appear in computations but are not detectable by experiment. It could be that they are examples of mathematics becoming reality. :chin:

Not necessarily. Imagine two dots drawn on a balloon, that is then inflated. The dots move apart exponentially as the angle from the radius increases — counterpunch

Hmm. Maybe. Think of yourself at the center (0,0) of a circle of radius R in the plane. One point on the circumference is at (R,0) and the other point is above that in the first quadrant. As the circle inflates the angle, A, between the points from your perspective is constant, but the radius increases as does the arc distance between points. The arc length between points is S=RA Thus the rate of change of S wrt time t is S'(t)=AR'(t). Now, if R'(t)=CS(t), you get exponential change in S(t).

I think there are vast more of those mathematicians who push the theorem to the sidelines close to the border of logic and logical inquiry and insist that it has nothing to do with anything else in the field of math than what the theorems state — ssu

Not quite. It has the respect of most in the math community, but most of those think they will never come up against that roadblock. Me included. It has arisen, however. For example Goodstein's Theorem.

{ } = the empty set = philosophy? :smile:

Something is still missing — magritte

Intuition, instinct, imagination, et al.

Then there is the issue of computer generated proofs. What kind of thinking is involved there? — magritte

I suspect ingenuity on the part of the program creator. After that a computer does what a computer does. Here's a good article: Computer Proofs

I think the reason the concept of smallness intrigues me is because some scientists talk of 'point particles' which are essentially one dimensional objects that occupy no space. — Proximate1

Make that zero dimensional.

but is wind similar to waves in the ocean — The Opposite

What happens at the altitude of the jet stream is highly influential on surface winds. There are wave patterns at that level, so that, for instance, a trough generates unsettled weather at the surface, with low pressure systems appearing. It's complicated trying to predict the behavior of the jet stream. Or it was many years ago when I was a meteorologist.

Proofs in mathematics are said to be discovered, as they are logical possibilities that arguably would exist even if no one discovered them — Janus

It's the theorem that's discovered/created first. Then the search for a proof. Math is not just challenging others to solve a stated problem, although for many that is a competitive aspect highly desirable.

Good luck on bringing him onboard. I gather it's been difficult to get professionals to participate in the forum. But if he has a lot of spare time he might. :smile:

I've been scanning and reading Bell's book that you kindly linked. Although I've been a math person for many years I've been concerned only with certain areas of classical complex analysis and a few abstract vector spaces. So, when I read that in smooth infinitesimal analysis (SIA) all functions from the reals to the reals are continuous I immediately think, what of a step function, like Heaviside function, how can that be interpreted as continuous? However, I gather that in accordance with the axioms of SIA such functions may not be considered in the first place, as they are not defined on the (augmented) reals. In other words, all functions defined in the system are continuous by definition. Am I correct? If so, then the claim that all functions from the reals to the reals are continuous is misleading. But perhaps I'm not interpreting things properly. :chin:

I like the infinitesimal line segment approach to continuity. It's how I program many of the functions I graph on the computer, although I'm using short real line segments.

Perhaps you and/or fishfry would comment on these ideas and elaborate on them to make SIA clearer. Sometimes in math very arcane subjects arise from elementary observations. Like reading a research paper in which simple motivation is left out, replaced by a series of odd looking lemmas leading to the proof of a theorem. :cool:

Even a circle has microscopically fractal texture — Enrique

I'm not sure how the technical definition of a fractal applies here. Explain what you mean, please.

The Numberphile video on Zeno's paradox expresses concern about what is at the end of an infinite series with no final term. The mathematician said he wanted a *physicists* to explain it to him — Gregory

He asks whether actual time and actual space can be divided according to the math of ZP.

Atalanta is walking from x=0 to x=1. What is the first non-zero coordinate that she walks to? I'd like to know how mathematical analysis solved this. — Ryan O'Connor

Zeno's paradox when seen from the perspective of pure mathematics is easily dealt with using the limit concept, but giving it an anthropomorphic twist makes it absurd. And the idea of a first non-zero coordinate shows a very limited knowledge of mathematics.

Would you agree a wise intellectual warrior should first know his enemy before striking?

Humans created the expression of 'time and space', but the motion of time and space itself exists regardless of human existence — Tombob

I assume you meant "notion" instead of "motion". Did you? Do you have any idea of what you are talking about? Just curious. :chin:

I see that you are confused about the most basic aspects of mathematics, language and reasoning. On certain points, your understanding is not even at the level of a six year old child. I'm offering you help here, though I doubt you'll take it in. — GrandMinnow

Quite a soap opera. Can't wait for the next episode. :lol:

And third, dt is a differential form. They don't explain these in calculus so that's why everyone's confused about them. Bottom line is they aren't numbers and they can't be zero OR nonzero. — fishfry

This may be a little high powered. The differential I-form is $f'(x)dx$ and it is common practice when teaching elementary calculus or even advanced calculus to simply assign the value
$dx=\Delta x$ for the independent variable. I used Thomas several times for calculus and Olmsted for advanced calculus. In both the authors explain that dx could be any real number, and then they assume it to be $\Delta x$. Of course, dx is originally an infinitesimal. It's not a big deal IMO.

The point-based view has reached its limit, it will never be able to solve Zeno's Paradox. — Ryan O'Connor

There is no problem. Mathematical analysis took care of that years ago. Only a few philosophers remain addicted to it. But this is a philosophy forum, so it's OK to quibble. :smile:

Jon, thanks for the link to Bell's work. My area was analysis, so I will enjoy reading it. :cool:

I certainly like the idea of constructivism in that it is necessary to construct a mathematical object to prove that it exists — Ryan O'Connor

It is a lot more satisfying to do this than to argue indirectly, IMO. I have told this little story before, but it bears repeating: There was a PhD math student who spent considerable time on his culminating research project creating a mathematical structure about a particular set of functions, until one day he was asked to provide an example of one of these functions. It turned out he was working with an empty set. :sad:

I really hope this was helpful — Gregory

Thanks. I would need a greater knowledge of philosophical theory to really understand. :smile:

Philosophy says that a point is a negation of a line — Gregory

Please clarify this.

I had my weekly chat with an old colleague (math prof) living in a retirement home today and brought the subject of instantaneous velocity up. He has his doubts about the existence, as do some of you, and for the same reasons. It doesn't bother me either way. I'll use the term since others will know what I am talking about: a certain mathematical limit.

My view is that actual infinity should not be permitted in math any more so that than it is permitted in physics, but that's just my view and it's contrary to contemporary math so I'm willing to leave it at that. — Ryan O'Connor

Not all contemporary math. In complex analysis one may move to the Riemann sphere and take the north pole as infinity, but I stick to the complex plane and use expressions like "unbounded" instead. Now, set theory is another animal altogether. Not my cup o' tea, but fishfry is an excellent in-house expert.

No, if "sqrt" represents an operation, then "sqrt(2)" represents that operation with a qualifier "(2)". — Metaphysician Undercover

This is absurd, but then you do see things from an unusual vantage point, inaccessible to many.