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.

From Britannica:

The Democratic Party has changed significantly during its more than two centuries of existence. During the 19th century the party supported or tolerated slavery, and it opposed civil rights reforms after the American Civil War in order to retain the support of Southern voters. By the mid-20th century it had undergone a dramatic ideological realignment and reinvented itself as a party supporting organized labor, the civil rights of minorities, and progressive reform

In a nutshell.

Do you think you understand Wittgenstein? — Joshs

From ChatGPT:

Overall, Wittgenstein’s profundity lies in his ability to challenge and expand our understanding of how language functions and how it shapes our experience of the world. His insights continue to provoke thought and debate, making his contributions to philosophy both deep and enduring.

and

Wittgenstein's ideas have influenced various contemporary philosophers and mathematicians who are interested in the foundations of mathematics, the nature of mathematical truth, and the philosophy of language. While his impact is more philosophical than technical, it has contributed significantly to ongoing discussions about the nature and practice of mathematics.

I admit, it's been sixty years since I have read anything by the man. At the time I was most interested in his impact on mathematics. However, this was about the time I was taking my one and only course in foundations (naive set theory), and was rapidly losing interest in the subject.

I am not a fan of Wittgenstein's philosophy as it seems to make common sense notions into philosophical "strokes of genius" — schopenhauer1

Ditto. A war hero, yes. Otherwise my eyes glaze over quickly. Early in my mathematical career I tried reading him but found little to interest me.

(2) There are no set theory experts in this thread (or, to my knowledge, posting in this forum). — TonesInDeepFreeze

That is all right. You are enough good to teach me a few things in set theory. — MoK

Yes, and are quite knowledgable in this regard. I never studied anything beyond naive set theory and I have appreciated reading their posts, both instructional (and at times, argumentative). As for reference books, I am compelled to mention one that I have found wonderfully informative and written by a colleague from Colorado College: Introduction to Topology and Modern Analysis (George Simmons)

Continuity is perhaps best approached through elementary topology. Here is what Simmons says:

In the portion of topology which deals with continuous curves and their properties, connectiveness is of great significance, for whatever else a continuous curve may be it is certainly a connected topological space.

Of course, in the case of the reals the previous discussion concerns connectiveness.

Do you think the cannon was a protest or a celebration? Presumably, it didn't have a ball, but was loaded blank? — Ludwig V

There were several Civil War cannons on the lawn of the old armory where the ROTC had made its home since before that conflict. I'm certain one fired a blank at that time. It's funny but I cannot find any reference to it in the media. But I think it would have been a celebration. Most people at the University disapproved of the Klan, and there had been some speculation the KKK might get ugly, but they backed off and were more or less silent.

I had an older cousin who lived in the country and was a member of the Klan. He had worked at some menial job and kept dogs for coon (racoon) hunting. His kitchen sink had a manual pump and chickens ran loose in his dirt yard. When I went out to visit him in 1963 with my fist wife I was astounded in the transformation. He now lived in a nice brick home and when we were met at the door he was neatly dressed and introduced us to one of his best friends, a black man who worked with him at the BFGoodrich plant nearby. The last time I had seen him was in the late 1940s when I was a child. The plant had opened in 1946 and he was still living the rustic life and a member of the Klan then.

Does an equation exemplify symmetry? — ucarr

More or less. Here is where you find reference to the term.

The second law of thermodynamics leads directly to Gódel’s Incompleteness.
Perhaps l should look at dark: matter_energy through this lens — ucarr

And off he goes, where he is led nobody knows. :smile:

Imagine a math space such that : 6+9 =/= 9+6; semi-symmetrical mirroring. — ucarr

A non-commutative operation. Lots of them in math. For example, function composition in general: f(g(z))=/=g(f(z)). Also called non-abelian.