"The principle of the excluded middle has only a scholastic and heuristic value, so that theorems that in their proof cannot avoid the use of this principle lack of any mathematical content." Brouwer

If things are purely finite, there is a smallest unit of space. That is impossible though because you just divide it further. If you can't it's zero and has nothing to do with the object.

1) Things are composed of infinite parts

2) For rigor, we can use Cantor's arguments for taking infinities out of infinities

3) So Banach-Tarski comes right from Cantor

4) Therefore you can make a universe out of a pebble

The conclusion is that the limits of calculus go out the window!

I've merely had the courage to take what Metaphysician Undercover is saying to it's logical conclusion

If you infinitely divide an object and line the pieces up largest to smallest, what is at the far end?

I think we can know what an orange is, and that it doesnt have less volume when cut in half. It can be so divided infintely, so it's infinite AND finite. This is so obvious

The infinite divisibility of objects, which even Aristotle admitted was real, means all objects are infinite within finite bounds. My claim is that this is a basic contradiction at the base of material reality and geometry. As in "God does not play dice", God does not create the world as a mathematician, but as a poet

If I were to to move to the bed, I have to move half that. If there is no half, then I am already there! There is NOTHING that stops this process. Calculus merely distracts people from seeing the contradiction at the very foot of geometry, the contradiction that makes geometry possible. Zeno took his reasoning further than perhaps even Parmenides would have, and confounded the Pythagoreans even further than irrational numbers did.

Godel says that mathematics is either contradictory or incomplete. I think it's clear that there is a contradiction in the first step of geometry

The Beatle song Here Comes the Sun has been analyzed over and over by music experts, but anyone can listen to it and get stuff out of it from their own minds. Likewise anyone can see that objects in the world are both finite and infinite. And it must be, because we are made of matter and there must be a contradiction in it. https://en.wikipedia.org/wiki/I_Am_a_Strange_Loop

Now people are saying objects have no size. Oh boy!

Bertrand Russell on the Axiom Of Choice: "At first it seems obvious, but the more you think about it the stranger the deductions from this axiom seem to become; in the end you cease to understand what is meant by it."

Zenos paradox shows the infinity within the finitude of objects. The fact we can break a candy bar in two shows this applies to our world. That is enough for banach tarski. You can take infinity out of infinity. The extra cantor stuff well extra

In Quantum Psychology by Robert Anton Wilson, he writes that we have a thought of ourselves, and a thought of that image, and on to infinity. Math can synthesize thoughts, but not matter

Math merely floats over Zeno's paradoxes because they are about logic, not math. Zeno is a shortcut to Banach Tarski

The axiom of choice results from the infinite divisibility of objects. The fact that the object is still finite is why banach tarski is a paradox.

One of Kant's antimonies (which means ideas that have only contradiction in them) is about this very subject

Mathematical equations don't really deal with this problem. This has to do with matter. Split a banana in two and the two halves equal the whole banana. The process of division goes on to infinity, so the banana is infinite and finite at the same time. That's Zeno (when not applied to motion, which has the same problem). Many is always many when it comes to size/volume. So the world is an Escher painting in the flesh. I don't see why Zeno's paradox is not a paradox but Banach-Tarski is. The latter flows directly from the former, and there is no BT without Zeno

Standing in between two mirrors, someone can easily see Aristotle was wrong about Zeno. Yet the mirrors are finite.

Irrational numbers are not only irrational, but they go on forever while spatially being finite. The same applies to pi and the circumference in the sense of infinite fintude. Maybe math proves that something can come from nothing since spatially finite comes directly with spatial infinity. Nothingness is not dark, but white and shining says the Tibetan Book of the dead

Numbers apply to the world. One microphone is not two. Points show that objects in the world are both infinite and finite. Escher paintings exist but what we are talking about is matter, not perception. The eternal halving of the hypotenuse shows there is no final term. The.Pythagorean theorem, when we say the smallest units are the right sides, leads to length being irrational in the hypotenuse. Contradictions everywhere

"The world is unlike unto itself.. the one is the sphere of appearance... We have to think pure flux, opposition within opposition itself, or Contradiction. For in the distinction, which is an internal distinction, the opposite is not only one of two factors- if it were so, it would not be an opposite, but a bare existent- it is the opposite of an opposite, or the other is itself directly and immediately present within it.... it is that world itself and its opposite in single unity. Only thus is it distinction as internal distinction; in other words, only then is it in the form of Infinity." Hegel, Phenomenology of Mind

To apply infinity to matter in the universe seems unavoidable. Irrational numbers too go on forever. So Aristotle is wrong. Non-organic things can only be defined by cohesion. A lamp may be one thing for us but another for another culture. If the lamp shade is not sufficiently "stuck" to the lamp, you have two objects. So nominalism is justified from every angle, contra Aristotle. Now, finite geometry has been a complete failure. Indeed, I don't know much math. But I don't feel like those who do truly empathize with the problem of something being spatially infinite and finite in the same respect. I'm fine with a round triangle, as long as others see my point that this is where math leads

Calculus tries to make "many" (understood spatially) into something else. So calculus seems to be in error. But does not my opinion on this lead to nominalism?

The debate in the late Middle Ages about how many angels can dance on the head of a pin was about this question, because angels are dimension-less. Nominalism was popular in those days too btw

1) http://philsci-archive.pitt.edu/2304/1/zeno_maths_review_metaphysics_alba_papa_grimaldi.pdf

2) If points are not zero dimensional, you can make a triangle out of it and half the hypotenuse would be smaller than a point!

3) Banach-Tarski paradox assumes that every object is the same size. Does not Cantor?

Hegel united hericlitus with parmenides. This is related to the math because volume deals with the world. Parmenides famously said you can't think nothing and that nothing is impotent because it is.no thing. But the East came up with zero and meditation where you can think nothing and feel it's power

"Complex analysis" (where complex means "having both real and imaginary parts") is hard to justify because we get our math from the world, where many is always many and not one.

Hegel had interesting things to say on limits. Google it.


"The linear series that in its movement marks the retrogressive steps in it by knots, but thence goes forward again in one linear stretch, is now, as it were, broken at these knots, these universal moments, and fall asunder into many lines, which, being bound together into a single bundle, combine at the same time symmetrically, so that the similar distinctions, in which each separately took shape within a sphere [Parmenides's One], meet again." Phenomenology of Mind

According to Nietzsche, Hegel "systematized the riddle" of being and nothingness thru teaching that all is "obscure, evolving, crepuscular, damp, and shrouded".

Marx wrote:

"First making the differentiation and then removing it therefore leads literally to nothing. The whole difficulty in understanding the differential operation (as in the negation of the negation generally) lies precisely in seeing how it differs from such a simple procedure and therefore leads to real results."

Negations of nothing!

"Marx recognized the differential equation as an ‘operative formula’ — ‘a strategy of action’ which, when it arises, constitutes a reversal of the differential process, since the ‘real’ algebraic processes then arise out of the symbolic operational equation, which originally itself arose out of a ‘real’ algebraic process... The German mathematician Gumbel led a team to decipher them [Marx's 1000 pages of mathematical manuscripts] and published a report in 1927 listing the wide range of subjects dealt with." marxist dot com

This might be relevant, since against we understand math through the world:

http://www.hawking.org.uk/godel-and-the-end-of-physics.html?fbclid=IwAR297vm3qpeViCnrXcGXBuRo-PXCEXIOcUiQxlFQWk1e20Xvu-e90P_OhrA

Russell talked about Zeno's paradoxes in lectures and in books. He said that they had extreme subtlety. Ironically, you have to have an asymmetry between the hemispheres in order to see this

According to Russell, it was Weierstrass who banished points from calculus. I don't see how it worked without them though. Weierstrass said a continuous function is a static completed unity

"It is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever" says Bertrand Russell on Zeno's arrow, this is simply "a plain statement of an elementary fact" he says. I see motion has have an impetus, so I would disagree. Motion IS a supertask though, which makes it a huge problem. A haunting one

“motion can be understood as the position occupied by an object in a
continuous series of points in a continuous series of instants.”; Bertrand
Russell, I Principi della Matematica, (Milano: Einaudi, 1963), 637.

Yet he also said points were non-sensical.

“It is just as impossible for anything to break forth from it [the One] as to break into it; with Parmenides as with Spinoza, there is no progress from being or absolute substance to the negative, to the finite.”; Hegel, Science of Logic, 94-95.

Math cannot solve this. Maybe philosophy can

To have no length is by definition to not exist in the realm of geometry, and also geometry obviously applies to the physical world. Russell said at the turn of the century points were banished from calculus. Now non-standard analysis now says "infinitesimals are inevitable". Google that last quotation

My dad, i, and my twin brother all independently thought of Zenos dichotomy before we read of him. Thoughts like that caused my mind strabismus, which is how I am now ale to speed read. TMI

Is God the nexus, or other dimensions (as in the other world's interpretation of QM)?? Zeno's stadium paradox, when applied to the smallest unit of space (which seems to be zero), needs to be connected to modern physics. For the latter, each object has its own time perspective (like a "noise when nobody listens"). The relative motion of the columns seems to imply a smaller unit of time than the smallest unit of time

If God created the world, he created a dualism or hybrid of finitude and infinity.

"If the fractional times decrease in a harmonic series, while the distances decrease geometrically, such as: 1/2 s for 1/2 m gain, 1/3 s for next 1/4 m gain, 1/4 s for next 1/8 m gain, 1/5 s for next 1/16 m gain, 1/6 s for next 1/32 m gain, the distances form a convergent series, but the times form a divergent series, the sum of which has no limit." Wikipedia

Zeno's argument is really about Aristotle's "common sense" understanding of the world. In calculus numbers are not added one at a time. The value/limit that the addition is approaching is what is determined. There is still an infinity in the equation because infinitesimal are the quintessence of smallness. The question seems to be whether "many is always many" or whether "many" can be something else. Aristotle was wrong to say that an object is potentially divisible but not actually so, because something contains the same volume whether divided or not. What if God divided this computer infinitely? How can it be spatially endless yet finite when I type on it? I think Devans99 has a point. Maybe Minkowski's lecture on "staircase wit" will help

"If we look back on what consciousness formerly took upon itself, and now takes upon itself, what is previously ascribed to the thing, and now ascribed as to it, we see that consiousness alternately makes itself, as well as the thing, into both a pure atomic manyless One and an Also resolved into independent constituted elements, materials, or matters. Conssciousness thus finds through this comparison that not only it's way of taking the truth contains the diverse moments of apprehension and returns upon itself, but that the truth itself, the thing, manifests itself in this twofold manner... ;in other words, the thing contains within itself it's opposite aspects of truth, a truth whose elements are antithesis to one another.. It's being One contradicts the diversity of has." Hegel is his first published work dealing with zeno

If a guy (like Hume did) said that force was an undefinable thing, can we really say he is making no claims about ontology?

The only authors i know are avicenna, aveoroes, and the writer of the incoherence of the philosophers

To try to get a conversation going...:

Hume replaced Allah in the Islamic occassionalist system with matter as the prime mover (following Hobbes's on dynamics). With an infinity of possible forces involved in anything, science seems to be rather occultic. Zeno, through history's Pyrhonnians, probably led to Spinoza. Kant turn Spinoza's God into the unknowable, and Marx and his followers demystified the world by taking out anything that wasn't based in principle on matter. Then came Einstein who said time starts from motion. And finally we have Hawking, who points out there is no eternity outside of this universe for God to even act in. What are we left with except waiting for regularities to change?

See David Braine's book

Absolute time would prove there has to be a God I think. I believe in absolute space but see no necessity to believe in absolute time