Analogously, in mathematics absolute infinity corresponds to interpreting the intensional description of a total function or algorithm as being synonymous with an exact limitless extension . . . — sime

Apart from Cantor's speculations I never came across this idea. It can't be applied to sets so it sits out there in an unattainable splendor, ignored by most in the profession.

His instructions for the Art of Dreaming worked the first time I tried it. So simple.

I learned my skill from Castaneda's The Art of Dreaming, a succinct source though interwoven with Mexican Indian folklore. There is a wide spectrum of lucid dreaming, from ordinary slap dash dreams where one is along for the ride to the spectacular experience if awakening in an alternate reality, with full intellect and agency, and the feeling of bringing about whatever one wills to be.

But, as I've mentioned before, even in the latter crystalline state one may be able to clearly see the letters in a word, but not understand the word. Reading a newspaper without comprehension.

It's not easy to talk with someone who has not had the experience, and convey the stunning impression it makes.

For that matter, as I've already asked, what is your definition of "absolute infinity"? — TonesInDeepFreeze

:up:

A philosopher sometimes isn't much of a thinker at all. Some philosophers dedicate their lives to scholarship only. Meaning they study and analyse the works of other's before them and/or critique contemporary works (be they standalone works, other commentaries or other scholarly works). — I like sushi

And that is not "thinking"?

Like mathematics, in modern philosophy it might be difficult to come up with truly original ideas.

What about fractals? These are infinite spaces that you can explore on a computer. Their instantiation in a computer realizes what was an abstract infinity. — hypericin

Fractals are generated by simply iterating certain complex functions. They are not "infinite spaces" but images on computer screens. The iteration process is finite, say n=1000. There is no abstract infinity other than implied replications of patterns like turtles all the way down. You never really get there.

. . . directly measuring the quotient of two arbitrarily small intervals Dy and Dx with respect to some observed function — sime

dx may be arbitrary, but dy depends on dx and is not arbitrary. That's where the observed function comes into play. Maybe you are talking about something else. You seem confused about these things. Or maybe I misinterpret.

the classical definition of df/dx with respect to the (ε, δ)-definition of a limit, can be practically interpreted by interpreting ε to be a potential infinitesimal, and δ as representing a random position on the x axis given the value of ε , which when applied to the function yield df and dx as potential infinitesimals, i.e. finite rational numbers, whose smallness is a priori unbounded. — sime

Non-standard analysis is what you are referencing here in a somewhat befuddled fashion. What is a potential infinitesimal?

I have had dreams on occasion which quite literally felt more real than reality; they evoke a feeling of pure, crystalline clarity of consciousness which is lacking in waking life. — Noble Dust

Yes. It's a remarkable experience. The imagery of normal reality is a compromise between what is incoming through the senses and internal processing. When the former is lacking and internal processing is fundamental great clarity is a result.

Not bad. :chin:

As a young climber Crowley had a wicked sense of humor. I've wondered from time to time if he made his career as the creator of Thelema with tongue in cheek.

I've had similar thoughts about Max Tegmark and his Mathematical Universe hypothesis.

Science and engineering continues to work with classical mathematics , as well as classical logic, due to their vagueness, simplicity and brevity as a junk logic for crudely expressing ideas . . . — sime

I'm willing to concede that my colleagues and I have produced mathematical contributions that are worthless, but calling classical mathematics "junk logic" and "crudely expressing ideas" is a ridiculous accusation. On the other hand, that may not be what you are saying. It's hard to work through some of your lengthy paragraphs. Probably just me.

Unless infinity is formally identified with a finite piece of syntax, whereupon becoming a circularly defined and empirically meaningless tautology, infinity cannot even be said to exist inside mathematics, let alone outside — sime

In the complex plane "infinity" is called "the point at infinity" and correlates directly with the north pole of the Riemann sphere - a specific point. But I've never used this concept.

However, the shortest path between any two points is a geodesic ... — 180 Proof

On a curved surface. Straight lines otherwise. We're talking 3D, not 4D space-time.

I assume it includes such things as . . . . . . — Fooloso4

The thread goes well beyond recognized perspectives into imaginative realms.

Do straight lines exist? And even if you traveled the earth forever you will see the same places more than once. — TiredThinker

Take another sip, my friend, then off to bed. :yawn:

Some do. — Fooloso4

I'm familiar only with Bergson and his debate with Einstein. The amateur, Peter Lynds, brought this into modern times with his controversial article about the nature of time in Foundations of Physics Letters in 2003. But he didn't question an underlying nature of time, only that it occurred in intervals and not instants.

I'm curious if academic philosophers get seriously involved with issues like

What's the big mystery about time?

?

To put it colloquially, it isn't possible to give an exhaustive account of what it means to obey a given rule, because a tabular definition of the said rule can never be finished, implying that the intended meaning of a rule is publicly under-determined. — sime

The word "exhaustive" has a meaning here that most mathematicians would ignore. "f(x)=2x for all x that are positive integers" pretty much says it all.

F(g,x) = g(x-1) + 2 If g(x-1) is defined, else — sime

You call F(g,x) a "functional". This is not the commonly accepted use of the expression among math people. A functional operates on a function and produces a (real or complex) number. Not another function. Definitions in CS may differ from those in math.

I guess the arguments on this thread are too subtle for me. And I admit I don't read the long posts carefully. What I see is starts of patterns that follow from some algorithm, like f(x)=2x for x positive integers. Then one asks, Are there other algorithms that produce the same existing pattern? (2,4,6,...) :(2,4,6,8,...) vs (2,4,6,7,...) e.g. The answer is yes. This is such an obvious conclusion. I've never encountered this sort of conundrum in my research. Usually one tries to show a well-defined pattern arising from some sort of process has a certain property, like convergence or divergence.

But math is so incredibly diverse I'm sure there are those in the profession that ponder such possibilities.

@fdrake is more up on modern math. Comments?

I rather like the idea of philosophy of a topic one is unaware of. So what would be step one?

You benefit from the fact that the moderators have a soft spot for fake scientific claptrap — T Clark

:rofl:

You've been watching Another Life on Netflix, right?

So what more do you want? — Bitter Crank

Don't stop now! :gasp:

That 5 is an increase from 3 ipso facto makes it an increase in value because it is that value being described by increase — InPitzotl

The charm of TPF :roll:

Question: Are Tegmark's ideas more entertaining than Cornwell 1's physics pornography? Or, are they more of the same? :chin:

I recommend "Ozark" instead. Riveting.

Doesn't sound bad to me. If he chooses a fair judge he'll pick up a few Republican votes as well.

Like this they fit around the Planck-sized mouth of the open 4d torus. They are tied to a closed 3d space, a sphere, that is wrapped around the 4d mouth like a 1d closed line, a circle, is wrapped around the mouth of a 2d torus. — Cornwell1

Is this as close to pornography as physics gets? :gasp:

sin(x) almost equal to x, ok but to start with? does this not depend on which units are used? — universeness

Sin(x) is a real number, not a degree; this is in radian measure. And it's true only near zero.

The three symbols you cite are used in a looser sense. I use them to mean approximately. The symbol in s(n) ~ t(n) means "behaves like", so its a bit loose also. Here's an example:


$\frac{n!}{{{n}^{n}}}\sim \sqrt{2\pi n}\cdot {{e}^{-n}}$

This is a form of Stirling's formula.

What about:

$\sin (x)\approx x,\text{ }\sin (x)\cong x\text{ }for\text{ }x\approx 0$

or the more mysterious $\operatorname{s}(n)\sim t(n)\text{ for large }n$

???

The latter is the least well-defined symbol. The former means approximately, which of course requires clarity.

Once one moves beyond 3D, higher dimensional "space" for me becomes an algebraic geometric concept rather than a reality. Very useful for predictions but that doesn't imply it truly exists.

The universe as a dynamical system certainly contains those critical points showing SDIC that create butterfly effects. At such points, in the continuum of time, the universe splits, with one trajectory prevailing. What of the alternate trajectories? Do they exist in alternate worlds, or simply as ghost-like speculations? Memories of non-events?

Tegmark is an interesting guy, but his ideas read like science fantasy.

As an ex-gymnast and climber, what you speak of is allowing the body to do what it does with no interference from other aspects of the mind. No thinking, just flow with the pattern, that pattern having become instinct with practice.

Intellectual activities are more difficult to analyze in this way. When I'm trying to solve a math problem I think about it to the exclusion of distractions, hours seeming like minutes. However, there is yet another dimension to this topic. As Einstein noted, taking one's mind off the subject and relaxing doing something else may allow the subconscious to send forth a solution or a new way of approaching the problem.

This effect is what hunts the philosophers of science. They are so caught up in inventing schemes and methodologies of science that they loose sight on its actual workings and are completely paralyzed when they actually engage in it. — Cornwell1

Are you speaking of philosophers of science or scientists themselves? Of course, frequently they are one and the same.

The relative God exists alright! — Agent Smith

The best among us is but human, although I would agree that Tom Brady is a God.

To equate the far end of a normal curve to a celestial being is beyond philosophy IMHO and a fantasy.

How can such ingenious structure exist by itself? — Cornwell1

Ours is not to wonder why, ours is but to do and die.

Geothermal energy does seem to have tremendous potential. There are about 100K people employed in the industry now according to Wiki. It's estimated that it would be viable as a primary source if customers were willing to pay a little more for energy. But in these times of massive inflation that's problematic.

And what's the difference with ≡, "identical to"? — Cornwell1

$2x=3-4x$
Conditional equation, true only for select values of x

${{\left( x+1 \right)}^{2}}\equiv {{x}^{2}}+2x+1$
Identity, true for all or nearly all values of x

The latter isn't used much anymore.

Perhaphs jgill, can offer better insight here, than I. — universeness

Nope. You did a great job, buddy :cool:

So I've decided to try and dip my toe in the water with programming. — Wayfarer

Good luck, my friend. :smile:

I'm not a scholar of Augustine's works, but I've yet to read anything that he wrote about philosophy that I would consider nonsense. — Wayfarer

"The good Christian should beware the mathematician and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of hell."

Poor old jgill. — Banno

Look upon my Icon and tremble, all ye of TPF !!

This thread has become nothing more than a conflation between verb and noun of the word "change"

I could find no code example — universeness

O0ps, sorry, I meant the icon is a product in the complex plane of a BASIC program I wrote . In this instance, the program created the unexpected demon from a coupled pair of differential equations: dz/dt=f(w,t) and dw/dt=g(z,t),where the functions involved contained sines and cosines.

During the time I worked I knew a number of mathematicians who would have little to do with computers. A bit surprising since numerical analysis was popular then, and some of them were actually researching in that topic. I did too, but computers were primitive and numerical analysis sought to speed up computations, even if the mathematician couldn't speak the CS language.

My initial encounter with computers was a graduate math course in numerical analysis taken in 1962. We wrote short programs, turned them in to someone behind a window where IBM cards would be punched, and finally after a day or so, run through a machine the size of a large room. Then we would find we had made a mistake, and would repeat the process over several days.

It was not a pleasant experience.

I know of some prominent philosophers of physics with physics or math degrees: David Albert, David Wallace, Huw Price, Dennis Dieks. — SophistiCat

Certainly doctorates in physics or math - especially theoretical physics - can place one in the borderlands of philosophy and science. That was my point. Thanks for the info.

I wrote many programs in my very early days as a teacher in BBC BASIC.
Having to number every code line was fun eh? — universeness

N=1/N was a trick question. Sorry :cool:

I now use Liberty Basic. I used Virtual basic until one morning I opened my computer and discovered that Microsoft had deleted the language. Over the years I have tried a number of languages, Pascal, Fortran, Mathematica, C++, etc. But it seems the more sophisticated they are, the more they cater to the popular applications in math. My interests are about as as far from popular as one can be.

I enjoy the challenge of programming a complicated and unusual math process from scratch. Click on my image to see an example.

I haven't had to use required numbering in some time. Neither VB nor LB require them.

Now how do I explain it? — Agent Smith

$\sum\limits_{k=1}^{n}{\left( 2k-1 \right)}=2\sum\limits_{k=1}^{n}{k}-n=2\frac{n(n+1)}{2}-n={{n}^{2}}$

Nighty nite, my friend

This thread illuminates what it is to be a philosopher in modern times — jgill

No, it doesn't. To see what philosophy in modern times looks like, read some actual philosophy, e.g. here: https://philpapers.org/browse/time/ — SophistiCat

Since we are talking about time, glancing over some of the papers in the metaphysics of time (the truly philosophical area) much concerns the A-theory and the B-theory, and reading on one finds that subjects in physics, like relativity theory, come into play. Here the philosopher can only use popular versions of physics phenomena in their arguments - unless they have more in-depth knowledge of physics.

In the philosophy of mathematics it would appear that one becomes equally versed in foundation and set theory in order to make contributions. But I speculate. You may be right.