**This Crap Is Not Even Wrong**

In reponse to my recent series on motion, people (mostly annoyingly boring mathematicians, ha ha) write to tell me that I am wrong to deny continuity (infinite divisibility) just because it leads to an infinite regress. They tell me that there is nothing wrong with infinity and that it is possible to traverse a continuous distance in finite time because, as you do (assume the thing that must be proved is correct and hope the other guy did not notice: priceless), the remaining time and distance diminish (they always forget to add,

*forever and ever,*at the end). As Wolfgang Pauli would say, this crap (emphasis mine) is not even wrong. In fact, any argument in favor of infinity is stupid to the extreme, that is, its fallacy approaches infinity (pun intended). I am experiencing a Zatoichi vision as I write (for context, see Physics: The Problem With Motion, Part I) but please read on.

**Infinite (Almost) Stupidity**

If one believes in infinity, the fallacy of one’s stupidity approaches infinity because, if infinity existed, anything and everything would be infinitely big compared to the infinitely small. But, wait, that's not all of it. Anything and everything would be infinitely small compared to the infinitely big. So there you have it. Infinity is a system in which everything is infinitely big and infinitely small at the same time.

Have an infinitely great weekend.

## 26 comments:

For something (anything) to be finite, it must exist within the context of a larger and less finite space- otherwise it does not have any measurable boundaries. This is also common sense and irrefutable logic (at least utilizing the concepts as they are currently defined).

While our universe may be finite, there is still something outside it that is larger, and something outside that as well, ad infinitum.

I'm going to outline the very simple proof that there is a need for a concept of infinity:

Suppose all sets are finite (this means no infinity). This implies the set of positive integers (Z+) is finite. So, let n be the cardinality of Z+. n is a non-negative integer since the set is finite. Furthermore, n + 1 must be a positive integer (hence, an element of Z+). But |Z+| = n implies max(Z+) = n implies (n+1) is not an element of Z+. Contradiction. Therefore an infinite set must exist.

Infinitesimals (the infinitely small) are relatively more difficult to work with but the concept was developed into rigorous theory through nonstandard analysis. This even gave Newton some trouble (he called them fluxions and did a lot of hand-waving at the time; Cauchy, among others, found infinitesimals weren't necessary for standard analysis) and wasn't formalized until the mid-twentieth century. However, you seem to be more interested by ideas of infinite sequences, convergence, and derivatives, which definitely do not need infinitesimals to work. Most any text on real analysis will clearly explain it.

Matthew,

Even if I agreed that there is something larger than the physical universe (quite possible), I don't see how that would mean that infinity is a valid concept. Is this what you're arguing? If not, I guess I don't understand what you're trying to say. Sorry.

Suppose all sets are finite (this means no infinity). This implies the set of positive integers (Z+) is finite. So, let n be the cardinality of Z+. n is a non-negative integer since the set is finite. Furthermore, n + 1 must be a positive integer (hence, an element of Z+). But |Z+| = n implies max(Z+) = n implies (n+1) is not an element of Z+. Contradiction. Therefore an infinite set must exist.See what I mean about annoyingly boring mathematicians? The above argument is infinitely wrong for the follwing reasons:

1) It completely ignores my refutation of infinity as if it did not even exist.

2) Numbers and symbols do not exist physically. They are abstract concepts, i.e., thoughts in the mind. The inability to distinguish between the abstract and the real is a common mistake among mathematicians.

3) It is incorrect that n + 1 must be a an element of Z+. Z+ is not infinite by definition and does not contain all integers as claimed.

Please think about what you write and refrain from making such stupid comments as I am not shy about pointing them out for what they are.

Let me attempt to rephrase:

The concept of finite space can only exist within the context of a larger more expansive finite space, meaning that there is no beginning nor end to the quantity of finite space required, in other words the succession is infinite and must remain infinite for the concept of finite space to even exist. You can't have one without the other.

1. A proof does not have to refute arguments made by others. All a proof must do is take logical steps from one statement to another. That is what makes it a proof. If two proofs give contradicting results then either one must be wrong or the first principles must be inconsistent. You did not give a proof so there's no inconsistency nor paradox to address. I could dissect your arguments and give a point by point analysis of where you fail to provide a decent proof (for example, unqualified assertions and unsubstantiated deduction) if you'd like, though.

2. I did not give proof for a physical manifestation of infinity, nor was that the point. The proof shows that infinity is a useful concept which maps to existing mathematical structures.

3. The proof assumes Z+ is finite and then finds this can not be true. This is also known as proof by contradiction or reductio ad absurdum. You claim that Z+ is not infinite by definition and does not contain all [positive] integers. If Z+ does not contain all positive integers then Z+ is not Z+ (since Z+ is defined to be the set of all positive integers). In other words, if you believe Z+ (I'll call this finite set Z'+ from now on to avoid confusion) does not contain all positive integers, then which positive integers is it missing? Why is it not possible to take the union of the set of positive integers that Z'+ lacks and Z'+ to get the set of all positive integers?

Let's suppose again that all mathematical entities must be finite. If this is so, consider the collection of all (finite) sets. The number of sets contained in the collection must also be finite by your claims (let's call this number n). This means we can take the union of all the sets in the collection (let's call it S). Since S is the finite union of finite sets, S must also be finite. Now take the power set of S, P(S). |P(S)| = 2^|S| > |S| >= n. So S is a set containing more sets than the total number of sets. How do you resolve this in your system?

Matthew wrote:

The concept of finite space can only exist within the context of a larger more expansive finite space, meaning that there is no beginning nor end to the quantity of finite space required, in other words the succession is infinite and must remain infinite for the concept of finite space to even exist. You can't have one without the other.I see. I think it's important to distinguish between something that is infinite and something that can grow indefinitely. The former is impossible while the latter can and does exist.

chzchzchz wrote:

I could dissect your arguments and give a point by point analysis of where you fail to provide a decent proof (for example, unqualified assertions and unsubstantiated deduction) if you'd like, though.By all means.

I did not give proof for a physical manifestation of infinity, nor was that the point. The proof shows that infinity is a useful concept which maps to existing mathematical structures.Useful only to mathematicians but not to physicists.

The proof assumes Z+ is finite and then finds this can not be true.Not at all. The proof assumes that Z+ is finite AND contains all integers. Then it finds that it doesn't. Where is the proof in that?

One more thing. I wrote:

Not at all. The proof assumes that Z+ is finite AND contains all integers. Then it finds that it doesn't. Where is the proof in that?In other words, since we already know that the the set of all integers cannot be finite, assuming that a finite set contains all integers is absurd.

All we know is that a set of all integers cannot be not finite. Your proof does not show that there can be such a thing as an infinite set. Having said that, anybody who likes to play what-if type games, is free to do so.

Here's my analysis of your "proof",

If one believes in infinity, the fallacy of one's stupidity approaches infinityad hominem

if infinity existed, anything and everything would be infinitely big compared to the infinitely smallThis mixes up infinite and infinitesimal. They're not the same thing (e.g. you can diverge into infinity by a simple unbounded strictly increasing sequence such as a_0 = 0, a_n = a_(n-1) + 1, but you converge to zero with sequences that get smaller in magnitude such as a_0 = 1, a_n = a_(n-1)/2 by virtue of being Cauchy and the completeness of the reals). Furthermore, the metric for comparison was never defined so I can only speculate why everything is "infinitely big compared to the infinitely small". For instance, I don't see how zero can be viewed as "infinitely big".

Anything and everything would be infinitely small compared to the infinitely big.This assumes some symmetric property here that's not clear. Also, what happens when you compare the "infinitely big" with the "infinitely big"? Likewise for the "infinitely small".

Infinity is a system in which everything is infinitely big and infinitely small at the same time.This seems to be trying to reach a contradiction here by claiming that nothing may be "infinitely big" and "infinitely small" at once (again, why is this inadmissable?). However, your previous statements took "comparisons" one at a time, not all at once. Here is what you're doing:

Let x be "anything".

compare(x, infinitesimal) = infinity

compare(x, infinity) = infinitesimal

compare(x, infinitesimal) != compare(x, infinity)

Contradiction (expected compare(x, y) == compare(x, z))

In other words, you expect your comparison function to be independent of the second variable. If the second variable does not influence the outcome of the evaluation of the function, it's not much of a comparison between two variables. On informal grounds, we could investigate the properties of this expectation and see that it falls apart as soon as we try to find an element that is "smaller" than 'x' and another element that is "bigger" than 'x'.

The proof assumes that Z+ is finite AND contains all integers. Then it finds that it doesn't. Where is the proof in that?The proof shows that if Z+ is finite, then it can not contain all positive integers (that's our contradiction). This leads to the conclusion that Z+ must be infinite. That's it. Now you may be saying there is no set that may contain all the integers, but then what are your criteria for a valid set? If a set may only be a set if it is finite, how many sets are there? If you claim finite, then I refer you to the power set argument I made earlier. If you say that the power set does not exist, then what are your set axioms?

I think it's important to distinguish between something that is infinite and something that can grow indefinitely. The former is impossible while the latter can and does exist.The reals can not be obtained by "growing indefinitely" (they are defined through the least upper bound axiom). This is the distinction between countably infinite and uncountably infinite.

Louis,

The proof presented by chzchzchz is based on the assumption that Z+ has two properties:

1) It contains all positive integers

2) It is finite

Then, it provides a logical sequence of steps where it generates a 'new' integer which is not contained within Z, thus requiring Z to be bigger than we originally defined it. This logical sequence of steps could be reproduced infinitely.

The logical conclusion is that one of the 2 properties we assumed is therefore false. Z+ can either contain all positive integers, or it cannot be finite.

If you chose Z+ to not contain all the positive integers, you are simply ignoring its most fundamental property, which is to contain all positive integers. If that property does not hold, then the set you are calling Z+ is no longer a true representation of the concept of Z+.

Therefore, the other property (which is the one being truly tested here) cannot be true, and Z+ must be infinite.

I see. I think it's important to distinguish between something that is infinite and something that can grow indefinitely. The former is impossible while the latter can and does exist.This is question i felt when I was age 9-10 that why a/zero = infinity; that someone told me. but, I felt that this is not Infinity, but indefinite.Every infinite series are indefinite until they are constrainted under some conditions. Like the way we think, every present AI problems are NP complete, until they are considered finite CSPs.

From My childhood I have many strong doubts about Continuity, definition of Integration given by Newton, and Ramanujan's Infinity problems. I will say them to you all soon. I am feeling very excited to write down them and giving their alternative solutions points that I have found.

chzchzchz,

I'm a little busy right now but I'll write an answer to your analysis as soon as I can. Let me come right out and say that there is no meat in it. You talk the talk but you can't walk the walk.

Ricardo Padilha wrote:

The logical conclusion is that one of the 2 properties we assumed is therefore false. Z+ can either contain all positive integers, or it cannot be finite.I agree with all that. But everybody understands this stuff intuitively. It's not rocket science.

Therefore, the other property (which is the one being truly tested here) cannot be true, and Z+ must be infinite.Not true. This is the sort of fallacy that comes from mathematicians wanting something to be true at all costs. They're not diligent when it comes to examining all their assumptions and end up deluding themselves and the rest of the world with them.

The proof only shows that Z+ cannot be BOTH finite and be a set of all positive integers. THAT IS ALL IT SHOWS.

It certainly does not show that an infinite set of positive integers is logical or possible. In other words, the viability of infinite sets does not follow from the "proof". That's complete BS. Sorry.

[I use bold typeface below for clarity]

As I made clear at the beginning of my post, the context of my post is the subject of continuity as it pertains to motion. I am at war with the idea that there can be such a thing as a continuous (infinitely divisible) line segment. So here goes.chzchzchz wrote:

Here's my analysis of your "proof",

If one believes in infinity, the fallacy of one's stupidity approaches infinityad hominem

Where is your sense of humor? This is a blog, not a textbook. I am deliberately disrespectful to authority because it serves my rebellious purposes. Readers must be willing to live with my chosen style or ignore my blog.if infinity existed, anything and everything would be infinitely big compared to the infinitely smallThis mixes up infinite and infinitesimal. They're not the same thing (e.g. you can diverge into infinity by a simple unbounded strictly increasing sequence such as a_0 = 0, a_n = a_(n-1) + 1, but you converge to zero with sequences that get smaller in magnitude such as a_0 = 1, a_n = a_(n-1)/2 by virtue of being Cauchy and the completeness of the reals).

This is a blog, not a math textbook. When I use the word infinity, I am referring to the everyday usage of the word. It’s a concept that encompasses the infinitely small and the infinitely big. An infinitely long and continuous line segment is an example of infinity.Furthermore, the metric for comparison was never defined so I can only speculate why everything is "infinitely big compared to the infinitely small". For instance, I don't see how zero can be viewed as "infinitely big".

Wow. Who said anything about zero? Zero has nothing to do with the continuity of an infinitely long line segment. Zero is not a thing. The comparison metric is simple and has to do with quantities. Since the concept of infinity assumes that both infinitely small and infinitely big quantities exist, the operation consists of comparing two quantities. You are free to represent those quantities with numbers, if you wish, but that would not affect the proof.Anything and everything would be infinitely small compared to the infinitely big.This assumes some symmetric property here that's not clear. Also, what happens when you compare the "infinitely big" with the "infinitely big"? Likewise for the "infinitely small".

You just found another contradiction. No comparison can be done.Infinity is a system in which everything is infinitely big and infinitely small at the same time.This seems to be trying to reach a contradiction here by claiming that nothing may be "infinitely big" and "infinitely small" at once (again, why is this inadmissable?). However, your previous statements took "comparisons" one at a time, not all at once.

Both comparisons are done at the same time. Why? Because the infinitely small and the infinitely big are assumed to exist simultaneously.PS. You may reply if you wish but I am tired of this exercise. I must be moving on to more interesting topics. Thanks for your input but, as you can see, I will never change my mind about the silliness of infinity. My view is that it is an abstract concept and not very useful as such. Mathematicians can have fun with it but it has nothing to do with physics, in my opinion.

HAHAHHA....oh! Louis, how did you read my mind ??? :-p

This lines you wrote here is in my mind over the 10 years.

About Zero, I believe "Zero is an Indefinitite content in mathematics". No one knows the volume, area, amount of zero. What we cannot define in visible volumes or units is called as Zero.

To the supporter of Newtonian Calculas, I just like to say one thing, that Newtonian definition of Integration is also Discrete in real sense. By the definition, Integration of f(x)dx is {f(x+h)- f(x)}/h;

Now the h is creating already the integration as summation of Discrete values. So, if we plot the f(x+h) and f(x) in Euclidian space, the h will be difference between the two points.How are they continuous then ?? That is the reason every Integration gives us "Error" values or additional Constant values.

Now, if we look the definition of displacement is also fraud and worng. If we look the definition of Work as Energy conservation hypothesis, the definition of displacement will become failed. For example, you have two roads for going to position A to B by your car. One road is having the smallest distance and another one is having the more distance. So, as a consumer of Petrol what will you think ? Going to A to B by the smallest one or larger one? Obviusly, the smallest one, because the consumption of Petrol will be less there or Energy will be used there less. But, what is the definition of the Work teaches us? That, the amount of work depends on the Displacement path only; but, the work originally depends on the path we take to go from A to B.

How long will the acedemic society bluffing us??????

Look this video, ArthurBenjamin already publicised the idea for changing the 21st century's mathematics education.

Tanmoy,

I agree with you that, contrary to what we've been taught, calculus is 100% discrete. In fact, everything is discrete. There is no exception. As I said in my series on motion, continuity is as stupid as it gets. It's the religion of cretins.

Thanks for your comments.

Louis, I'm not going to bother asking why you are side stepping a rigorous definition of comparison or why my version of your "proof" is wrong since you seem to be immediately revolted by proof-oriented mathematics. Instead, it's time to remind you that you are incredibly close-minded; you claim nothing will change your views. You should not be proud of deliberate ignorance. Rather than using precise definitions, you appeal to intuition. Unfortunately, evidence points to the universe being anything but. Otherwise, why did it take so long to develop physics in the first place? Clearly since the phenomena of static electricity was known to the Greeks, they should have been able to use pure intuition to develop a theory of electromagnetism. Precise definitions are required for communication of precise ideas. Relying on intuition makes for little more than sloppy conversation.In the spirit of being open-minded, I'll offer what evidence will change my mind in your favor: give a verifiable result that the current system can't provide. In this sense you must show where there is a problem in the system and then give a solution. You need actual equations here showing what is wrong, not just pictures and invective. This will show you have a clear understanding of the current state of physics and can, in your words, "talk the talk". Once you establish the problem in existing terms, it's time to give a solution that not only fixes the problem but also coincides with experimental data. That's it. As it stands, there are plenty of uses for infinity in physics (the path integral formulation of quantum mechanics comes to mind). Here are some things you can do: predict the mass of the fundamental particles to a high degree of accuracy (or even better, find a tigher bound on the Higgs boson), find how gravity works at quantum scale, and so on. Pretty much any novel result will do. Physicists already observed situations where classical physics (e.g. schoolboy physics) doesn't work and have compensated accordingly, so you're going to have to go beyond trying to disprove an already discredited theory.

Tanmoy, I'm not sure what you mean by AI problems being NP-complete until they are cast as finite CSPs. An NP-complete problem will always be NP-complete under a solution preserving polynomial time transformation. This is why NP?=P is such an important problem. You could use the same polynomial time transformation to make all NP-complete problems solvable in polynomial time on a deterministic Turing machine. For example, 3SAT can be trivially cast as a finite CSP but it will still be NP-complete (although this does not preclude the possibility of the existence of a polynomial time solution). If you're willing to give this more thought than Louis would, I'd be more than happy to discuss it with you over email (just append an @gmail.com ...) instead of hijacking Louis's blog for now off-topic discussion.Newton did not give a rigorous definition of integration, so it's not worth talking about. In fact, it seems you're confusing a limit formula for differentiation with integration; this is disasterous. You're starting off with a flawed premise, so it's not worth going further. Riemann, Stieltjes, and Lebesgue were instrumental in giving a proper definition of integration. Reproducing the definition for the Riemann integral requires some machinery that takes a while to build up, so I'll omit it. Wikipedia explains it, albeit rather poorly. Again, I can correct any mistaken notions you may have about calculus over email.

You should not be proud of deliberate ignorance.You're the ignorant one, my friend, for ignoring the fact that your so-called proof that infinity exists does not even come close to showing anything other than the simple observation (already known to everybody with two neurons between their ears) that a finite set cannot contain all possitive integers. Well, whup-dee-do!

The jest of it is that infinity and continuity are soundly falsified for the simple reason that they lead to an infinite regress. It's so simple a child will understand it.

Feyarabend was right when, speaking of scientists, he wrote in Against Method that "the most stupid procedures and the most laughable result in their domain is surrounded with an aura of excellence." I call it crap in a jewelry box.

chzchzchz wrote:

In the spirit of being open-minded, I'll offer what evidence will change my mind in your favor: give a verifiable result that the current system can't provide. In this sense you must show where there is a problem in the system and then give a solution. You need actual equations here showing what is wrong, not just pictures and invective. This will show you have a clear understanding of the current state of physics and can, in your words, "talk the talk". Once you establish the problem in existing terms, it's time to give a solution that not only fixes the problem but also coincides with experimental data. That's it.I am not out to convince the physics community of anything. My peers are the world. I already showed where current physics is wrong by not accounting for causality when it comes to motion. It's a simple observation that anybody without a political agenda can understand. It is not rocket science. I don't need to stand on my head and do a neutron dance for your amusement.

Equations are less than a dime a dozen. The only thing that will knock everybody's socks off is an actual physical demonstration. Floating a 100-ton block of stone in the air would be very nice. That will come soon enough. Good luck with your infinite series.

Newton did not give a rigorous definition of integration, so it's not worth talking about.Ok. I was wrong for just only mentioning the Newton's name.

In fact, it seems you're confusing a limit formula for differentiation with integration; this is disasterous.Yes. I am confused by the Integration long time back, and using Integration method is always created errors in my programmes. Thats why I hate the

reductionism and Equations.What will you like ?? "Errorless but new" or "Error-prone but classical"? Choise is upon you.

You're starting off with a flawed premise, so it's not worth going further.What is flawed ??? Using Newton's name as a Frontpage of Calculas ? or Showing, Why is reductionism giving poor-errorprone results? Or why definition of Integratation is discrete series , but not for continuous series?

This is not clear to me.

I'd be more than happy to discuss it with you over email (just append an @gmail.com ...) instead of hijacking Louis's blog for now off-topic discussion.Thanks for your kind interest, I really would like. But, why in chat at Gmail? Open Blog is always better for freedom and free discussions.

PS: The main problem is Human brain does not know "how to delearn from memory?" This cold Mindset to the old one is main problem to accept anything new.

chzchzchz:

welcome to tanmoydeb06@gmail.com without any hesitation.

btw, my timedomain is GMT+5.30

Louis, I now realise that logic will not persuade you and you rather rely on your instincts. So I'll try to formulate a thought experiment that will hopefully appeal to your instincts:

Let's say you have some orbital cannon that can shoot stuff into space at extreme velocities. The stuff this cannon shoots travels on and on, in straight lines if not influenced by gravity (or atmospheric resistance, which does not apply anyway in space). Now, I understand that this might conflict with your view of motion, but all my experience agrees, and so does pretty much everyone else's. Take the Voyager probes as examples, if nothing else. So I am hoping this side-steps your motion theory, as the method of "propulsion" (or inertia) is not under question.

So, you use Hubble or some future super-duper telescope to find regions of space where there is nothing and where nothing will cross the the future trajectory of your projectile when it's there. [If necessary, imagine you're omniscient and just know where everything is and will be.] You then fire lots of projectiles in all directions towards empty space.

What will happen to those projectiles? If you don't allow infinity, will they just at some point hit the "wall of existence" and stop? To me, this sort of a wall is an unnecessary invention for which there is no proof. We have no evidence for it, so why do you need to invent it to satisfy your intuition?

Quoting you: "An infinitely long and continuous line segment is an example of infinity."

Is the trajectory of these projectiles not exactly that: an infinite line segment. If not, where does it end and why?

If you are unhappy with projectiles, then consider photons.

You still do not explain why infinity or continuity lead *necessarily* to an infinite regress. Just because it does in the way you think about it, doesn't mean your thinking is only possible way of thinking. Maybe it could be, God forbid, wrong. So please share your logic.

I'll try one more approach, quoting:

"If infinity existed, anything and everything would be infinitely big compared to the infinitely small. But, wait, that's not all of it. Anything and everything would be infinitely small compared to the infinitely big."

This is simply not true. Say I take a grain of sand, measuring 1mm across. It might seem superficially infinitely small when viewing from the scale of the whole universe, or something bigger than the universe, or something even bigger than that (ad infinitum). But it is NOT in fact infinitely small. It remains finitely small, measuring 1mm, even if that *seems* infinitesimal at your grand scale and grand things you compare it with.

Likewise the bigger-than-the-universe-thing (ad infinitum) is still the size it is, even if you compare it with and electron, or sub-sub-sub-quark-like-thing or whatever. One thing might *seem* infinitely big or small in comparison, yet they are not and remain the size that they are.

Even if there is nothing beyond the sizes science currently can observe, this does not invalidate the infinite scale. Just because they scale goes to infinity, doesn't mean that actual objects do.

Just because there might be something really, really small, and something even smaller than that, and something smaller than that ad infinitum, doesn't mean all these things don't have finite sizes.

[I'm no physicist, but I guess this relates to why black holes have no size and are called singularities; their only properties are hypothesised to be mass, charge, and angular momentum. They are no longer objects in the traditional sense. They have NO size.]

I am repeating myself, I know. I just trying to make sure you follow.

I think it's important to distinguish between something that is infinite and something that can grow indefinitely.Thanks for this bit of insight, it really got me thinking in a whole new way.

I'm not sure if it's exactly what you meant, but I started to visualized various linked things expanding and contracting indefinitely. It's an intriguing idea.

It's far to easy to forget that words like infinity are just made up.. really nothing more than a theoretical interpretation of something we don't yet understand.

Language becomes the context within which

weexist. But just like finite space, it is nothing more than a foundation on which we build greater understanding, forever expanding, but never exceeding itself..A real-world example of infinite distance:

continuous travel around earth :>

If you were a two-dimensional creature, living on the surface of a sphere.. you'd see that your 2-D world is infinite.

Zdravko said...

A real-world example of infinite distance:continuous travel around earth :>

Why are u trying to argument something, which you not want to understand yourself, but saying, what did your teacher tell you at school???

If you were a two-dimensional creature, living on the surface of a sphere.. you'd see that your 2-D world is infinite.Again saying this is not infinite but Indefinite. Because, you dont know in Numeric figure how much will that be to represent that?

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