However, my primary focus is on the objects themselves, such as the Cartesian coordinate system. I believe this system needs a parts-from-whole, continuum-based reinterpretation, as the current understanding relies heavily on the notion of actual infinity. — keystone

Not sure what you mean by actual infinity. Are you speaking of infinity as a sort of number that can be arithmetically manipulated, or infinity as unboundedness? I have always used the concept of the latter rather than the former. But set theorists use both I think. Please provide an instance of "actual infinity" in the Euclidean plane. A projection onto a sphere is not allowed.

Sounds interesting. Life in the complex plane. By the way have you seen much of the modern graphing software that's so good at representing complex functions and Riemann surfaces and the like? Don't you wish you'd had that back in the day? I wish they'd had LaTeX, I always had bad writing. — fishfry

No I haven't. And I have little interest in Riemann surfaces. I have used MathType for years with Microsoft Word for writing purposes. For imaging, I have found BASIC is excellent for what I want to do, and have written many math programs. The image of the Quantum Bug on my info page was done with a simple program. Higher, more sophisticated languages seem to be directed toward what is popular in math, and what I do is virtually unknown.

Reminds me of Tim Gowers's distinction between problem solvers and theory builders. — fishfry

Somewhat similar, but not quite the same thing. I've never particularly enjoyed solving problems, but rather exploring where certain specific ideas in classical analysis lead. The celestial aviators can cruise the heavens taking us ground troops on ethereal adventures.

Thus the make up isn't at all close to the natural 50/50 divide, hence mathematics is male dominated. — ssu

Here's a personal anecdote that may be telling: My PhD class had several women. One dropped out for health reasons, and another was the top student, by far. Shortly after graduation she married a forest ranger and became a housewife.

In the international research clique I joined there were several women, but more men than women. A fairly close colleague, a European woman who had left behind a role as housewife, became the holder of an endowed chair at a major Scandinavian university.

wouldn't this mean then that mathematics is different from being just a social construct of our time? — ssu

Mathematics can be thought of as a structure or system or whatever, but certainly it is not "just a social construct". Sociologists are a little irritating.

Yet today, set theory is a basic part of the undergraduate math major curriculum — fishfry

Perhaps at the more prestigious schools, and maybe at less elite institutions as well. I think I checked on this for Harvard and found such a course, but the branch of the state university where I taught doesn't seem to offer such a course, although set theory is mentioned in a couple of conglomerate classes.

Math is very much a social process. — fishfry

Absolutely. Although many mathematicians work alone much of the time, with social contact allowing critiques by colleagues. It gets very social when one publishes a paper, with regard to referees.

Isomorphisms have everything to do with structuralism. An isomorphism says that two things are the same that are manifestly not the same. That's structuralism — fishfry

I might comment that whereas isomorphisms are very important in mathematics, not all practitioners are heavily involved with them. I have written many papers and notes without mentioning the word. However, the trend for the past how many years, maybe 70 or 100 or so, has been to rise above the nitty gritty of much of classical material and look for generalities that show how one subject in one area is "isomorphic" (use here in a more general sense) to another subject in another area. Or create generalities that when applied specifically to a lower level collection of results show them to be instances of one higher outcome.

I didn't go in this "modern" direction, and my late advisor would speculate that where he and I explored (ground troops, not aviators) would at some future time return to fashion. These days there is such a plethora of subject matter I'm not sure what is fashionable.

So, I am really really interested in whether or not mathematical objects exist in a mind-independent way. Would there be numbers even if we weren't here? I want to say yes. I think that numbers would be here — Pneumenon

Was it Einstein that said something to the affect that God gave us the natural numbers, all else of mathematics are mans' ? Or something like that. When I think of one of my theorems about the indifferent fixed point of an infinite composition of LFTs that converge to a parabolic LFT I never stop and wonder if all that preexists in a timeless realm beyond the thoughts on man. I do believe it exists in a potential way.

a societal phenomenon and a power play that a group of people (read men) do — ssu

In America women make up 25-30% of PhD students. 15-20% of math faculties. Not entirely men.

More like that the truths in mathematics are tautologies: a statement that is true by necessity or by virtue of its logical form. Wouldn't that description fit to mathematics? — ssu

To me this seems like a word game. Describing a theorem, if A then B, requires specific terms, going beyond its "logical form". The idea of categorizing math as a tautology contributes nothing to its practice.

when it's just what mathematicians declare doing. — ssu

Years ago the maid for a prominent mathematician was asked what her employer did. She replied, after some thought, "He scribbles on pieces of paper, grumbles, then wads the papers up and throws them in the trashcan."

perhaps the old idea of math being a tautology comes to mind — ssu

Statements devoid of content? (Frege)
I think not.

The only reason you don't want math to fully apply to reality is because you suspect a problem with infinite divisibility, right? — Gregory

I've never known a fellow mathematician who would have agreed with this. A mathematical philosopher perhaps. Let's see what @fishfry has to say. I always enjoy his commentaries.

Perhaps a very stupid question: why isn't Math referred simply to being a system? — ssu

Math itself is full of systems, like the system of the natural numbers, or the system of addition. So, in a sense, you might say it is a system itself, but a very complex system. @fishfry will give a much more sophisticated answer to your question.

If I had to guess where you are headed, I might say that taking a continuum (a line,say) as axiomatic somehow you are cutting it into a fine mesh using the S-B Tree. But how this has a bearing to elementary calculus is a bit foggy. Perhaps Farey sequences to partition Riemann integrals. Just guessing.

Why is it that the intro to calculus/analysis textbooks I’ve read never mention topology? — keystone

Over the years colleges have designed their curricula to suit the levels of abstract thought students can bring to the classroom. Calculus is taught in high schools at minimal levels of sophistication and during the first year or two at college with a bit more rigor. But even there the emphasis is on understanding applications of the subject. Most students in these classes are not math majors. Analytic geometry is part of this instruction. The student progresses to a higher level at the junior or senior years. At this point they are usually capable of the sort of abstract thinking that underlies calculus in an advanced calculus course, or an introduction to real analysis course. As for topology, I taught an introductory course at the junior/senior levels.

All of the above may not be true at a sophisticated university with high entrance standards.

And the above are generalizations. Individual students may be more or less proficient than I have indicated. For example, a number of years ago a young freshman at my school registered for my advanced calculus course - and received an A (he was also a talented climber and we became friends). Then I have had seniors who barely passed.

(I began college at Georgia Tech in 1954, and was fortunate to be one of the few incoming students who scored high enough on the entrance exam for me to start anywhere I wished, so I was placed in an experimental class of beginning calculus, immediately taught with epsilon-delta precision, with rigorous proofs. For the first half of the semester I had hardly a clue what was going on, while some of my classmates seemed to understand the material. Then halfway through all of it suddenly made sense. After that introduction, when I got into the regular curriculum for the next semester it seemed almost trivial)

I will be interested in what younger and more agile brains make of this. :chin:

How is that any less of a leap than starting with curves, which are inherently continuous? — keystone

And then straighten out a (continuous) curve and you have a continuum, which the OP argues does not exist. Perhaps you should start a thread entitled "The Continuum does exist".

I've never been able to see where it is you are going. Maybe it's just me, old and weary of days.

It works fine for me — Art48

When it searches for Adamford.com it goes to your site, but on and off for adamford.com - as you have listed it - it goes to car dealerships. And Google stopped me twice when I wrote www.adamford.com . Then not. Peculiar. Nice site you have though.

My emotional involvement is because the Bible tells enormous lies about God — Art48

How can you be so certain?

The validity of Christianity was once a philosophical topic. Are you saying it's been excluded from modern philosophy? — Art48

True. Long ago. It should be excluded. My opinion.

Continuous object: In 1D, the proposed fundamental objects are of two types: (1) open-ended curves, which are inherently continuous — keystone

So we begin by defining such curves as "inherently continuous". That seems to solve the problem. Why proceed? Why dabble with sets of points that may fill up a curve - or not?
Just chop up such curves and there we are. Bend them a bit to go to two or more dimensions.

From the desk of an old mathematician.

For me potency means a function I design that when implemented by choosing a point from its domain produces a really big number.

For me potential means infinity as simply an unbounded process as in "goes to infinity".

For other math people infinity may be a term described by an axiom.

And it is these ignorant people, the most uneducated, the most unimaginative, the most unthinking among us, who would make themselves the guides and leaders of us all; who would force their feeble and childish beliefs on us; who would invade our schools and libraries and homes. I personally resent it bitterly. — Art48

When I see a diatribe like this I speculate why its author is so vehement. Why does your website on your bio page list www.adamford.com, a site Google warns against as a scam, or, when going to adamford.com, is a car dealership device?

It's true that backwoods churches overreach at times, but I grew up in the Southern Baptist tradition, going to large, big city churches in which the sermons were at times quite sophisticated. The congregations were largely professionals who would have scoffed at the examples you cite. In the academic environment in which I worked a number of colleagues were Christians and attended church.

Obviously, you have an emotional involvement in this issue. But I don't see it as a modern philosophical topic. But that's just me. Others here may differ.

A continuum is a decomposition of a hyperspace module with sidewise-partitionable step-wise radii — TonesInDeepFreeze

And you say you are not a mathematician! :cool:

"The standards society adopted" are largely unexamined. It is a card house of assumptions and I'm challenging a specific one — Benkei

Therefore, affirmative action or equal opportunity initiatives would be justified to help these individuals reach their potential — Benkei

This could mean adopting hiring practices that prioritize diversity, ensuring that supply chains are free from discrimination, and promoting workplace cultures that are inclusive and supportive of all employees. — Benkei

I have seen up close a corruption of Affirmative Action, and whereas I had thought it reasonable before, afterwards I was reluctant to support it. It is a quota system in disguise.

The entire point of q2<2 is to define that set without reference to irrationals. — fishfry

You've made your point. Don't rub it in.

For a moment I was thinking q^2<2 normally is q<sqr(2) for positive q, but if irrationals do not exist this inequality is invalid. It seemed to disconnect at the sqr(2), which doesn't exist. Fuzzy thinking. Lets move on.

OK. I used to teach elementary point set topology occasionally but it has been over a quarter century ago. Best for me to avoid this discussion at my age.

Ok. You are disconnecting Q at a point that does not exist in Q. Thought you were restricting all points to Q. Usual approach to this is to assume the underlying reals.

I did convince myself that if you take the rationals by themselves, you can define a topology by all the "open" intervals (p,q) with p and q rationa — fishfry

Yes, usually it is inherited from the usual topology on the reals. But ignoring the non-reals seems OK. Looks like it is connected as well. But not a linear continuum since it doesn't have the LUB property. Rusty here I"m afraid.

What is your point? What do you think? — I like sushi

Different issue I think — Benkei

Yes, not quite what the OP is all about. This is an example of the sort of legal issue that might ensue upon the death of the wage earner, in which a will is contested, or the allocation of part of a pension payment plan upon retirement. Normally, marriage would be involved.

In this case there have been no legal issues raised. As to whether the wage earner here was morally entitled to the money he received, I don't see why not.

When I ask myself if the money I earned while working was "appropriate", I can only say it was determined by the standards society adopted in a competitive environment. There was no absolute moral factor invoked.

I find the issue becomes more or less about what an individual can do and what others believe they should do — I like sushi

Two men are friends over a long period of time. #one has a regular job that provides a decent retirement plan. #two is artistic but has little interest in planning financially for the future. At retirement age #one retires on his pension. #two then asks #one to share his pension with him. #one refuses.

Did #one make the right moral choice? (this actually happened)

Assuming this statement is true, what do you think is its philosophical significance? — 180 Proof

:up:

Isn't the subconscious process deterministic? Doubts are not allowed in a deterministic system. — MoK

For me, the word "doubt" applies to a conscious state, not a subconscious state.

I think it is good you are getting back into the discussion. Who knows what might come out of this thread? My only reservation - and ignore if you like - is to perhaps not bring up the Stern–Brocot tree.

The subconscious process cannot resolve the conflict when we have doubt in a situation. That is true since the options are real when we have doubts and we don't have any reason to choose one option over another option — MoK

I don't believe we are aware of all the information that enters our mind. If that is the case what the subconscious processes may indeed inform us - in what seems to be an act of free will.

:up:

If I recall correctly, another poster mentioned point-free geometry. — keystone

Instead of points one works with lattices of open sets. I don't see this as improving the intuitive understanding of continua. Continuity in elementary topological spaces rests upon the idea of connectedness. The topology of the reals is fairly well established, so maybe start by studying this.

What Cantor was or wasn't is not particularly relevant. Just my opinion.

You are traveling through a maze and reach a fork. Here you experience a maximum degree of doubt (uncertainty), and the consequences of making a wrong decision are large. You take out a coin and toss it, heads to the right and tails to the left. The coin toss makes the decision. This is hardly an instance of free will, other than deciding to leave the decision to the coin.

You are traveling through a maze and reach a fork. Here you experience a maximum degree of doubt (uncertainty), and the consequences of making a wrong decision are large. Now you ponder and then make a decision. Is this free will? Or does some internal neural mechanism in your subconscious "toss a coin"?

Only a special infinity can subsume the whole of math — Gregory

This sounds more theological than math foundational.

As a complex analysis guy you use the hypothetical point at infinity of the Riemann sphere all the time, don't you? — fishfry

Only on rare occasion. Normally, Infinity for me means unbounded. I don't work on the Riemann sphere. Yes, projective stuff is there in the background, like circles with infinite radius are lines, etc. But I am very old fashioned. Here is the sort of thing that has interested me.

If only the standard analysis of the reals had been discussed, with infinity not a member and infinitesimals not (re)introduced, perhaps things would have terminated long ago. Maybe not, but I would guess most physicists don't dabble in non-standard analysis nor are they concerned with the roles of ultrafilters in pointless topology. I could be wrong but even introducing ordinals into the discussion opens a Pandoras Box. Just my opinion.

Well, the Zeno paradox certainly threatens mathematics, especially the continuum concept. — MoK

Oh please :roll: While the word "continuum" is everywhere on this thread, what is really the heart of the subject is "connectedness" of sets. And the Zeno paradox does not threaten mathematics.

.

Anyway, the idea of someone, who doesn't understand that the set of natural numbers is not a member of itself, trying to grapple with how ultrafilters play into proving the existence of hyperreals is ridiculous — TonesInDeepFreeze

:up:

So the mid-20th century party is the one you joined? — Ludwig V

I probably registered as Democrat when I voted for JFK. I didn't give it any thought at the time. LBJ began his political life in the Senate staunchly against civil rights legislation, but reversed his position as the tides began to turn.

Do you agree with fishfry about what the party has become? — Ludwig V

More or less. Frankly, I don't know what it has become since the "Squad" gained influence.