↪jgill
Not sure i followed that, but isn't it also correct that the length of the "hole" is zero? — Hanover

The "length" of any number on R is zero. Numbers are positions on the real line, designated as points, none of which have any "length".

(To see if you comprehend what I say, Counselor, you might show that the interval [0,1] is in 1:1 correspondence with the interval [0,1). But don't worry about it.)

There is a nice mathematical way to cash our the intuition the original poster is gesturing towards. See The continuum as a final coalgebra shows that the real numbers (a.k.a. the continuum) can be constructed from infinite steaming interactions over infinite sequences of natural numbers. — FirecrystalScribe

Category theory. Beyond my grasp and interest. Nevertheless, what is an "infinite steaming interaction"?

Is this the same as saying that the infinity of all integers is larger than the infinity of all even integers? Or, is it the same as saying that that is you have two sets, one composed of all numbers and the other composed of all numbers except the number 3, the first set is larger than the second?

In my first question, both sets are countable.

In my second question, neither are countable because both contain irrational numbers — Hanover

Regarding question 2, consider two horizontal non-negative real lines, one above the other. The line on top is missing the number 3. Do they have the same cardinality? That means a one-to-one correspondence between the two lines. Start at the bottom line at 0 <-> 0 on the top line. Let x be on bottom line and y on top line. Then x=y until the "hole" where y=3. You have a 3 on the bottom, but not on the top. So you introduce the correspondence 3 <-> 4, then it's as usual, x=y, until you reach 4 on top, which has been taken. So, stipulate 4 <-> 5. Since this pattern persists indefinitely the one-to-one correspondence holds forever.

I've made the comment on several occasions that the Schrödinger equation, in its simplest form, is a concept from elementary calculus: the instantaneous rate of change of something is proportional to the amount of that thing at that instant. In calculus the solution of such an equation involves the exponential function, and in the setting of complex analysis, e^it = cos(t)+isin(t), which is a wave structure. Not a physical wave. Nice video, thanks.

Does my complicated odyssey end should I go deep enough within myself to open the door of my inner Conscience, then find a way to step into my soul? — jufa

I like your poetic style, but unless you describe this inner progression in at least minimal detail and elaborate on "complicated odyssey" the reader is left with nothing but style.

An experiment I read of some time back involved a person reacting to a warning signal without actually being conscious of it. If you are reading a book you are aware of the act, though your thoughts are focused on what you are reading.

Sounds like the Peace Corps here in the USA.

In mathematics, a theory has substance when it is deemed important or significant in some way by a community of scholars.

First the commitment to non-contradiction, to both avoid contradicting oneself in belief — boethius

Do you mean blocking the ability to see both sides of an issue? Give some examples please. I don't read lengthy essays.

But when scientists go beyond compiling facts to explaining their significance, they are straying into metaphysics, and doing Philosophy — Gnomon

Not if they speculate within the normal scope of science. But, if they conjecture that action at a distance has religious connotations, or that the universe is a reification of mathematics, then, yes.

. . . but everytime you open a banana, you are the first person to ever see it. — Hanover

How exciting. Be sure and post its image on TPF. :cool:

↪Quk
I've never seen anything uncaused. I have no reason to think that would fail prior to the big bang. Maybe a better thing would be to say "I want to know why the singularity existed" — AmadeusD

Good point. Years ago I published a mathematical result that, more or less in this context and under certain conditions, could be interpreted showing that the further back in time one goes from a current event the less it matters what the starting point is. This assumes no boundaries on what lengths the causal chain extends backward.

The Big Bang seems a bit like an essential singularity in complex analysis, as does a black hole. Absolutely bizarre things happen in its vicinity.

Howz bout dis'...if it ain't yer original thought, or yo' original thought about somebody else's thought, don't post? — alleybear

I wonder how many original thoughts have been posted on this forum? Fifty years ago my math advisor said, "You might think you have an original thought, but it's likely someone has had that thought before, perhaps in another context".

I believe the "American Dream" is more along the lines of starting life with little and making a financial success and/or gaining recognition later on. I have been privileged to know someone who started out almost penniless in his early twenties (I once went with him searching for stovepipe in trash dumps) , and became a billionaire, and was on an advisory panel with Bill Clinton in the 1990s. And I knew another of that generation who started from poverty and eventually created a clothing line and was worth hundreds of millions of dollars when he passed a few years ago.

Are there other nations where this kind of Dream is possible?

Horatio Alger personified and wrote of this in the 1800s.

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: