Tuesday, February 10, 2009

Proof of Infinite Regression's Fallacy

The starting guess is that infinite regression is a contradiction, and like all contradictions assuming it is true results in finding that you can use it to prove anything. (From the book Zero, if 1=0, Winston Churchill is a carrot.)

This turns out the be the case, though in a somewhat interesting manner. For even one infinite regression to work you must already know that every possible statement is true. Because, the only way to work an infinite regression is to have an infinitely receding line of different statements. The only alternative is to repeat itself, which reduces the regression to petitio principii.

Petitio Principii
I learned something analyzing it in detail. Logic has something of a anisotropic crystalline structure. Something like a lightning bolt as well. Starting from any beginning assumption, you can use the laws of logic to travel down the forks and prove the endpoints, but you can't move back up, except insofar as you can infer the thunderhead from the scorch marks on the ground. "Ah, for this scorchmark, we have to assume this particular lightning bolt. Thus, this particular lightning bolt must have been true so as to prove this scorch mark." And so, even though the endpoint necessarily requires the premises, you cannot use the endpoint to prove the premises. Said another way, you can go backwards to learn, but not to prove, the premises. In yet another way, causes make effects, and while we can infer the causes from the effects, the causes do not flow from the effects.

Because every true fact must be consistent with every other true fact, you get not just one branching tree but an entire series, forming a nice anisotropic crystal; you can go down, but not up.

So any infinite regression that repeats itself has exactly the structure of petitio principii, and tries to have a cause, a premise, flow from an effect. This is simply not the way logic works.

Ergo, to be a true infinite regression, it must not repeat itself.

Ze Turtles
La Wik says this; "There would never be adequate support for P1, because the infinite sequence needed to provide such support could not be completed."

This is incomplete. It begs the question, actually. In fact, what I need to prove is exactly that the sequence cannot be completed. The reason it cannot be completed is that it must contradict itself.

Luckily, the proof has already been done for me. Like the formula for pi will eventually spit out any sequence of digits you want, a proof using infinite regression will eventually need to state every possible statement. Or, equivalently, given an infinite amount of time, every single possible event will occur not just once, but an infinite number of times. As a result, either every single possible statement needs to be true, instantly showing that it's a contradiction because it proves everything, or else you can't complete the sequence without contradicting yourself after a finite number of steps. And yes, actually, I can show this using the epsilon/delta definition, if it isn't obvious.

Also, I just realized, to fully complete an infinite regression always leads to a circular argument, because 'every possible expression' includes the proposition you're trying to prove.

I will happily show how any example you want to propose fits into one of these categories. (Or all of them.)

To be an expert in philosophy, you need to be able to just sit down and form these arguments, which is exactly what I did earlier today. "You know, it's time to prove the fallacy of infinite regression." So I worked out what infinite regression actually means, by eliminating everything it can't mean, and then worked out the consequences of that definition. And poof! Knowledge appears.

Regardless of all this, infinite regression would be a sublimely useless technique. We will never accumulate an infinite amount of data with which to verify an infinite number of premises. I would have fallen back on this if I couldn't have proven that it was a contradiction.


Trying to figure out how to classify premises so I can check two possibilities.
First; given any single assumption, there are a finite number of other assumptions that are neither a consequence nor contradictory.
Second; assuming there are in fact an infinite number of possible assumptions, can I categorize them like 'even' and 'odd' to filter some out yet leave an infinity?

It seems extremely unlikely that there are an infinite number of distinct, non-contradictory, yet relatable premises. For instance you can assume infinite dimensions, and then you can have an infinite number of non-contradictory independent assumptions, and even relate them with the behavior in dimension x+1 being a function of dimension x. It's infinite, it regresses, and in fact you actually need an infinite number of premises to fully define motion in that space. {Even if it's x=1 and f(x)=(x+1)=0, and so x+2=f(x+1)=0, and so on.}

Right, so if you see any universes made up of infinite dimensions, the proof won't apply to them.

So, now, can you get an infinite number of premises in one dimension? In other words, can you make an infinite number of independent, relatable, non-contradictory statements about the natural numbers? No, you cannot. The natural numbers are entirely defined by the Peano axioms; everything we know about them are consequences of those. The same reasoning applies up to the real numbers, though of course complex numbers are two-dimensional and require at the very least assuming a second dimension.

So this isn't really a proof...yet. My hurdles are at once extremely high (dealing with infinity) and yet also low (they appear extremely different than anything we see empirically.) On the other hand, the exact same situation in reverse obtains for anyone who wants to use infinite regression to prove anything.

The ultimate question then, is, how many dimensions does truth have? Actually you need to start by defining dimension for premises and logics that aren't strictly numerical...

So, uh, someone wanna get on that?


Anonymous said...

Wow, this is a great post. I've been pondering infinite regression, myself, because I have heard it asserted countless times that an infinite regression is not logical yet pretty much anyone that makes that claim is quoting a philospher who is quoting a philosopher who is quoting a philosopher who is quoting a ....

Alrenous said...

I'm glad you like it, because that makes at least one of us.

Anonymous said...

You assume that just because a regression is infinite, that every possible configuration must occur an infinite amount of times.

This is not necessarily the case.

The string of numbers -1, -2, -3, ... and so on, regressing by 1 infinitely, does not contain each number an infinite amount of times, but is a valid example of an infinite regression.

Alrenous said...

You may be correct, but your proof is flawed.

The series of negative numbers contains a very finite amount of information. For example, I just exactly described it in a finite sentence.

AlexZy said...

Please, help me to figure out. Are you arguing against the possibility of infinite regression or for the impossibility to prove anything via IR? My English is not so good, so I could miss something in your reasoning, but I partially expected to find some keys to skeptics IR argument against the possibility of justification. Excuse me if this was not your concern.

Alrenous said...

I'm arguing against both; the physical possibility and the proofs trying to use it.

I'm happy to try to make things clearer for non-native speakers, though I suspect I don't have the necessary skill to succeed.

I can now state my argument better, I think. To prove anything, it must be written down in a finite number of words, which means a finite number of premises, which means finite information.

A truly infinite proof would require infinite information. Since information has mass, physics cannot contain anything that would require infinite information to specify.

Any proof by infinite regression cannot be meaningfully infinite, which means it either terminates or goes circular. If it terminates, it isn't a regression either, and if it goes circular it is just a fallacy.

AlexZy said...

Thank You for the reply. Please forgive my trivial notes below. I need them to define, whether I understand You properly. I am afraid, one problem is that there is no consensus about the so-called deflationary theory of truth and meaning in philosophy. Do 'meta'-statements provide extra information via predicating truth-value or meaning to statements about the world, or they are informationally equal? If '"The snow is white" is true' tells something distinct from 'The snow is white' (for me it is intuitively right), then there is an infinite possibility to produce new statements:

p; p is true; 'p is true' true etc.

To get closer to the skeptical argument, let's express it in a way below:

(1) Q is true, because it is justified by P.
(2) Q is true, because P' justifies that P shows the evidence for Q... etc.

Do You argue that there is a countable number of ways to justify the existence of or explicate relations between statements, objects and, the most important, statements about statements? I believe that infinity appears due to the need of a new theory of justification for each level or step of justification: P justifies Q, because it corresponds to facts and implies Q; 'correspond to facts' means '...' or take place when ..., 'imply' means '...' etc.

So, it looks like in my example (which is obviously not mine, but a variation on the old one) an infinite regression is supported by implementing an iterating operation to changing matter of statement. This iteration relates to form, so there seems no circularity in content presented.

Sorry for this long note.

Alrenous said...

I'm not bothered by length or language struggles. Instead I'm happy we share an interest. I go long myself - it would be hypocritical for me to object to long comments.

The meta-statement is ambiguous. It can be meant literally, or it can have its implied, second-level meaning.

If I say the snow is white, then I'm communicating that I believe it is white. (Nobody's burning coal nearby, say.) If I add that it is true, I'm just wasting time.

If a second person adds that it is true, they're communicating that they agree with my assessment. They can also say, "Yes, the snow is white," or, "The snow is white," and similar things.

I am indifferent to consensus. With one caveat, the deflationary theory is true - reality's is the only opinion I much care about. (Note that is it bad grammar to simply assert 'deflationary theory' though I can simply assert 'snow is white.' Although logicians often assert that P. Logicians and I agree that grammar doesn't make sense.)

The caveat being that a statement that fully defines the question may still need computation to get from the statement to prediction. Pi can be defined very quickly but there's no shortcut to knowing the 10,000th digit, if you need it for something.

Get at deflation from another angle. If I'm doing an experiment to determine whether, p, snow is white, do I need an additional experiment to show that '"snow is white" is true'? I don't think I do. I record a broad spectrum reflected from the snow, and I'm done - the whole chain of self-asserting statements is justified.

This angle method can be generalized. If the belief were in an impenetrable black box, can you tell the difference between one world where deflationary theory is true and one where it is false? I can't. I can't see any consequences except to itself, so it dies to the hard form of Occam's Razor.

The snow is white is true is true etc. chain is a good example of what I mean by a finite proof. We both understand what we're talking about - which means it was communicated in a finite time. Its apparent endlessness is illusory, as the actual information inside must fit into a few natural-language sentences.

Rather than talking about ways to justify, I'd like to talk about the way any particular statement is related to its justification.

My claim is that after stripping illusory infinities from justifications, all true statements have finite justifications or else go circular.

AlexZy said...

So, let's check if I got your general point properly. I will try to summarize it in my dummy terms. Please do not hesitate to correct me.

Putting it in physical terms, there are no infinite energies in the world, thus, due to energy-information interrelationship, every event brings a finite quantity of information as well. No infinite information - no infinite judgment and/or sequence of premises. The informational finiteness leads to tautologies (circular reasoning), whenever reasoning about a matter terminates or exhausts itself. I can accept that or something similar but more finely formulated.

If this grotesque interpretation was reliable at all, I would like to know how one should deal with the Gödel's formalization theorem?

Alrenous said...

The use of the word 'grotesque,' as with similar phrases, reliably indicates no communication is possible.

I conclude no communication is possible.

I will now stop pretending communication is possible.

Good day sir.

AlexZy said...

I called my interpretation grotesque only because I worry about my English. I really hardly imagine how a native speaker sees my expressions. Regrettable if this is the reason for stopping the communication.

Best regards.

Anonymous said...

In saying that the person who used an infinite sequence of negative numbers may be correct, yet their proof is flawed, how does this correctness effect your argument? Also, if you are appealing to logic and philosophy, you have to consider Munchhausen's Trilemma, correct? This Trlemma would suggest that infinite dimensions must exist for reality to exist, because otherwise you are left with circularity, or you allow that Churchill is a carrot. This is especially frustrating if you focus on the need for infinite regression in time, and the origin of existence itself. Eventually I had to allow for 1=0 but pretend that it could not be allowed to maintain a functional reality :-P

Alrenous said...

Re: infinite negatives, the key word is 'may.'

They may have the right answer, but they failed to demonstrate it. I do not believe in things I cannot personally demonstrate.

In addition, I've spent enough time on this already as compared to the likely payoffs, according to my judgment, so I'm not going to try to fix their proof for them.

So I'm trying to say I think their line of inquiry doesn't have any obvious fatal flaws, and might be worth pursuing. That said, I thought I made it apparent enough I'm not going to personally pursue it.


Re: Munchausen trilemma, I don't find the axiomatic solution unsatisfying. Instead, I find it obvious. Of course there are bedrock facts which neither admit nor need justification. That being the case, we approximate the actual bedrock facts by assuming certain more-convenient axioms for the question under inspection.

Or: do not pretend to question in your philosophy what you do not question in your heart.

Also, good luck finding a way to question things like A=A.

This seems to want an example. So, if I'm questioning dopamine, I take as axiomatic certain bits of chemistry, then reason forward to the effects on the mind. If I later find a need to question chemistry, I retreat the axioms only slightly, then reason forward. (Then recurse.)

Tangenetially, this made easier because I can safely rely on my habit of questioning the questionable. No matter how confidently I assert some axioms, I will easily and naturally un-accept them if I come across a good reason to.

Anonymous said...

Doesn't being able to follow a cause to an effect essentially allow an affect to be tracked back to a cause? If a=b, then b=a?

The only way one would not be able to fully track an effect to a cause would be if one did not incorporate all actual causes into the effect. In other words, one cannot give a definitive answer from incomplete information. Therefore, one must be able to assume with complete information, one can draw a definite conclusion.

Can we create a word that actually only exists within the realm of an idea? If I write the word 'unicorn', does that mean unicorns exist in the same way that horses exist? In other words, can we create ideas that exist only within the realm of ideas without being able to translate into a multi-dimensional realm? Would not such a thing as an infinite proof fall into such a category, and thus attempting to equate it with finite proofs contained within the 4th dimensional sphere actually be incorrect?

Alrenous said...

I agree. I don't think I looked at it quite that way before.

However, that established, if logic were calculus, there could be an infinite number of distinct implications between one physical entity and another. Or at least, that's how many seem to think.

Really just wanted to clarify that, though. It gets busted by the same principle. It's only meaningfully infinite if it requires an infinite amount of information to specify. Information has mass. We can be 100% certain there's no infinite masses to observe.

Ergo, logic isn't like calculus. (Or calculus isn't properly understood.)


If I were in your position, I would appreciate being told that I went overboard on the question marks.