Yet then it's typical that the limit approaches infinity — ssu

"x goes to positive infinity" simply means "positive x increases without bound". No need for the word "infinity". Simply shorthand. But, set theory is another matter and postulates all sorts of things.

I belong to a past generation who didn't venture beyond the Peano axioms. These days searching for relationships between the broad spectrum of math subjects is in fashion. I cheer them on from the sidelines. :cool:

There wouldn't be this kind of over and over repeating debate Zeno's paradoxes, if we fully would understand the infinite or infinity — ssu

Or the limit concept.

OK, if we have countable and uncountable infinities, what is the relationship between these two infinite sets? — ssu

Take the rationals in [0,1] and form the union with the irrationals in [0,1] and you get a continuum. They are complementary in the complete interval - which itself is a complete metric space with the usual metric.

The precision of position and momentum are proportional to each other such that a greater precision of position results in a lesser precision of momentum. — Moliere

How does this enter into a discussion of these Zeno type paradoxes? Define the momentum of a point as it progresses to zero. Does the tortoise have momentum? Too much of a stretch for me.

Is this in any way motivated by the uncertainty principle? — Moliere

Actual measurements fail beyond Planck's constants. These paradoxes are all hypothetical involving motions of dimensionless points along rational number scales.

(The proof of a limit is intensional, whereas the empirical concept of motion is extensional). — sime

Heuristics may not be precise, but its value can be substantial in an introductory course. I've never come across this sort of philosophical detail in elementary calculus, which includes lots of motion. "f(x) approaches . . . as x approaches . . ." is common language in math. But, whatever you say.

Hence calculus does not say that f(x) moves towards L as x moves towards p. — sime

Rubbish. :roll:

But also homeownership, which I kind of “fell for” in a way. Or the way living at home with one’s parents is viewed as being a loser — which ties into encouraging owning a home or renting. — Mikie

And what's the alternative? Are you implying that owning a home is a bad thing? Or renting?

I think it's fair to say that 'field' is used in many contexts: different disciplines in science and the humanities are commonly referred to as fields — Janus

Even mathematics is a little sloppy in this regard. A vector field is not a mathematical field. The reason I prefer the expression vector space. And if that vector space changes values at each point over time it is a time dependent vector space (or field).

So the question for you is, does every point in that field have a mathematical description, as do the points within physical fields? And if not, does that disqualify its description as ‘a field’? — Wayfarer

Field as a mathematical term or field as an area of land devoted to growing crops? Or field as an encompassing environment of some sort, a philosophical notion. Spacetime is not a math field, but contains various entities like magnetic fields that can be represented as math fields.

"Understanding" quantum theory means following the math, as Feynman said. Perhaps that is true of time as well. The math of relativity theory weaves an astounding vision far beyond what we might have imagined. If one entertains Tegmark's speculations that the universe is a mathematical structure, then time is one also. A reification of mathematics.

. . . excitation of the one universal field of subjectivity. — Wayfarer

From math to woo. A little like the aether.

As for understanding space/time, my Corgi still cannot comprehend simple high school algebra. We have to learn our limitations.

Are you recommending adding this to a classical education?

How to isolate an instant? Take a photo. — jgill

As I've explained above, that is an arbitrarily created "instant". So it provides nothing toward proving that real time consists of a succession of instants — Metaphysician Undercover

I would be surprised if there were a proof to the contrary. Isn't all of non-analytic philosophy speculation?

I don't know if you've had much interaction with the sometime contributor here, Apokrisis, but he has a lot of interesting things to say about biosemiotics, a field I didn't even know existed until he came along. — Wayfarer

I think he has a PhD in biophysics. This thread seems to be in a rut of sorts. He might add something original to the discussion. My own ideas, shallow as they are, is that time certainly exists and is a continuum of instances, like the points on the real line. How to isolate an instant? Take a photo.

Is it intended to be used for research purposes? — javi2541997

It's useful in physics for wave functions that might otherwise be cluttered with sines and cosines. It crops up from time to time in things that interest me. For example, the DE that defines a particular contour

$\frac{dz}{dt}=iz$ , solving for $z(t)$ where $t:0\to 1$

Then
$z(t)={{z}_{0}}{{e}^{it}}$
${{z}_{0}}{{e}^{it}}={{z}_{0}}\left( \cos t+i\sin t \right)=\left( {{x}_{0}}\cos t-{{y}_{0}}\sin t \right)+i\left( {{y}_{0}}\cos t+{{x}_{0}}\sin t \right)$

Euler's Formula: 1740 Considering the complex plane a vector space helps to see the connections.

$e^{i x} = \cos x + i \sin x,$

Neither space nor time come equipped with intrinsic measurements. Relativity sees to that. Without objects in play there is nothing. There is no independent space were it not for at least two objects. Then there is time, but only if those objects move through space. Space and time are the empty stage, coming alive only with actors therein.

But is this really a dream though? It doesn't sound like you were even asleep, if you noticed yourself strolling by a table, and you could even knock on the table to confirm that you were not asleep. — Metaphysician Undercover

Guess where I was walking? In the bedroom towards the closed door - which I walked through, like moving through a panel of smoke. I was fully conscious. (delightful experience, by the way)

But the question is, how can the subconscious so thoroughly deceive the conscious, so that the conscious doesn't even know that it's not awake when the subconscious is producing dreams — Metaphysician Undercover

In the sort of lucid dream I described, one realizes exactly what is happening. I remember testing the state by knocking on a table while strolling by, feeling the fibers of the carpet beneath my feet.

How is this possible, that my mind can allow itself to go into a completely distinct reality (which is not reality, yet I believe it to be reality at the time)? How is it possible, — Metaphysician Undercover

The closer one gets to lucid dreaming the more pronounced this effect. I've commented before on Castaneda's Art of Dreaming, and how the process leads to an awakening into an alternate form of reality, more vivid and compelling than normal, allowing one to become pure will.

The key to accessing this experience is summoning a form of consciousness while in the hypnagogic state. Perhaps this is a portal through which one can exert some control of elements of the subconscious, for the subconscious interprets sensory input.

The moment of coexistence of the breaking and unbrokenness is the actual breaking in unbrokenness. Physics and math have no ability to see it or describe it. — Corvus

Sure they do. A simple graph describes the aging of the glass, then, abruptly, there is a discontinuity when the glass breaks. Draw your own picture.

Now I see why fdrake retired as moderator.

Math can describe the motions and movements of objects in numbers and functions. But they are not time itself, is it? — Corvus

There is a continuity of existence that is mechanically measurable. A car sitting by the curb ages a bit over twenty four hours in a an approximation of an ideal or mathematical continuity. Unless, for instance, someone comes along and blows it up. Then there is a discontinuity of existence and the end of a mathematical parallel description. The Riemann integral concept in math analysis embodies the notion of addition of a sequence of temporal points, the distances between points shrinking to zero. When applied properly to dynamical systems that are analogous to physical change, predictions result.

The accuracy achieved is, of course, ultimately governed by Planck dimensions. So, whether Bergson's interval description of the atoms of time is cited or a "continuity" of points is described, is irrelevant.

I agree with everything you say. Latin used to be a high school course available to those students having aspirations in the medical areas, especially MD prep. That made sense. On the other hand, learning Greek could possibly lead to better understandings of ancient wisdom - which seems a bit superfluous in modern times. As a retired mathematician my familiarity with Greek didn't extend beyond a very useful alphabet.

Being perceived is not what it is for something to exist — Banno

A breathe of fresh air. A history over time exists whether it is recorded through human perception or not. Paleontologists discover this truth frequently.

For those who suspect math underpins the character of nature, then the passage of time might well be understood in mathematical rather than philosophical discourse. What does the limit concept say about time? In the ever expanding galaxy of mathematical subjects does time arise?

This is news to me. Hillsdale is a conservative college frequently cited in the more conservative news shows, and I have no problem with that. However most K12 schools are public, charter or non-charter, and where I live we hear little about Christian schools, although they exist here.

The boom of which you speak is the mouse's squeak, but loud, not weak. :cool:

↪jgill
What do you think about TREE(3)? — Arcane Sandwich

I think nothing of TREE(3). From my background in classical complex analysis it is merely notation without any connection to my area of interests. Just another of more than 30K math entries on Wikipedia.

One: It has multiple meanings. One such meaning is: It means "1".
Two: It means "2".
Three: It means "3". — Arcane Sandwich

Now I see why you seem infatuated with Graham's Number. I looked it up and found that it's the largest number associated with an actual mathematical problem (in Ramsey Theory). I also read of Knuth Arrows. New to me and a universe away from my mathematical interests. But whatever spins your wheels.

I see from your location on your info page you have followed Yogi's advice and have taken the fork in the road. Sorry to see the only moderator who is a mathematician leave that position. Fair sailing.

The problem with Time dilation is that it is another hypotheses i.e. possibility if you could fly in the speed of light. Could you fly in the speed of light? Could anyone? Even if you did, the result is not confirmed. It is a hypotheses — Corvus

https://en.wikipedia.org/wiki/Experimental_testing_of_time_dilation

I see nothing of substance in this philosophical discussion of time. But, if something can be physically manipulated and scientifically measured, I wager it exists. Time dilation does just that.

Do we think that DOGE will go after enormously expensive health care spending, which first and foremost is expensive because corporations make profit from it? — ssu

Interesting comment. I've wondered about our health care system, and this past year I have discovered how well it functions for senior citizens during the treatments of a broken leg and cancer at age 87, with complications. It's been one year since the fall shattering my right femur, then, in hospital, finding I had cancer elsewhere. I have a Medicare Advantage plan provided by my public employee's retirement program, and pay into it monthly, but my out of pocket charges were virtually minute.

A good friend recently had triple bypass surgery, and his surgeon told him that at the age of 82 had he lived in Ireland he would not have had surgery and would have been sent home to die.

More the dimensionality of thought than geometry. You might illustrate by describing an educational curriculum in elementary physics embodying your concepts.

:up:

It's very simple to show that infinite sets are not atl the same size — Janus

A misuse of the word "size".

Category theory would be the philosophers companion here, but uh... we haven't been trained in category theory in school or in the university. That is really something lacking! — ssu

This has come up before. There are categories in my own subject of complex analysis, but in order to work with them you need a solid background of complex analysis at the beginning grad level. Now set theory can start literally at the bottom and work up. I've mentioned before my intro to the Peano axioms and beginning at 0 and ending (at the end of the course) with exponential functions.

Category theory seems to be more a graduate school offering, whereas set theory can be presented at a much lower level. However, "New Math" of the 1960s and 1970s flopped when this was tried. Feynman was very critical of the effort.

. . . so one could describe the situation like mathematicians have outsourced the philosophical problem to set theorists — ssu

I like this. However, category theory - which includes categories of sets - an outgrowth of algebraic topology and what ever else of similar abstraction seem to have gotten into the game.

I was fortunate that the large state university I chose to get my PhD over half a century ago had a perfectly adequate but not elite math faculty, and I was able to do my research in a subject arising from classical complex analysis. Had I been confronted with category theory or a similar abstract topic I probably would have switched to computer science or electrical engineering.

Complex analysis, itself, has apparently moved up the inevitable steps of abstraction to the point that the arXive.org collections of papers on the subject are unreadable to me.

Classical mathematics doesn't need "infinity" as a sort of number with associated features. The limit concept works well. However, modern math defines the term(s) and goes into abstracta.

From the OP article:

My position is first that mathematics is an exercise in pure logic. It is not a human construct.

I'm not sure how to interpret this statement. If math is an exercise in pure logic and pure logic is a human construct, then so is math. So, is pure logic not a human construct?