All mathematical statements are tautologies.

All mathematical* statements are tautologies.

This statement, along with its’ many variations such as mathematics only produces tautologies, is categorically false.

I know that many of the readers are sophisticated so let me apologize up front for stating the obvious. (In fact the arguments that I am about to give are pretty much old hat, with the possible exception of some personal perspective)

Definition: B is a tautology of A if and only if (A implies B) AND (B implies A)

The reader should notice that if B is a tautology of A then A is a Tautology of B.

From a philosophic point of view, tautologies are generally viewed pejoratively because the statements are considered redundant and therefore no meaning will pass with the reference to B. (In the world of Mathematics tautologies can be great for linking together two separate theorems that may not appear to have anything in common. If you have read the book “Fermat’s Enigma” by Simon Singh, Simon Singit, and John Lynch you will recognize the importance of this comment).

Finding a counter example to the statement “all mathematical statements are tautologies” is easy. For example:

If x = 5 then x is greater than 3. Note that if x is greater than 3 then x = 5 is not a true statement.

Similarly, if x = 5 then x is a prime number. Note that if x is a prime number then x = 5 is not a true sentence.

Unfortunately, virtually every person that seriously addresses this subject has to undo the wrong that philosopher Ludwig Wittgenstein (a biography is interesting and morbid) has done.

Wittgenstein invented his own private definition** (private language is another subject) where he changed the meaning of the word tautology.

Originally he said that mathematical statements are tautologies because the conclusions are implied by logic. Latter (after realizing that removing the parallel postulate yielded different viable geometries) that statement was modified to claim that, in a given axiomatic system, all mathematical statements are tautologies.

In normal usage Wittgenstein’s sentences, similar to “all mathematical statements are tautologies”, should be replaced by “all mathematical statements are implied by their associated axioms”. The meanings are vastly different. I believe he kept the misleading pejorative terminology because he felt that mathematical theorems had no more content than the underlying axioms.

So, if I say all mathematical statements are implied by their associated axioms, what do we gain? The answer is almost, but not quite***, nothing.

What we loose is understanding and insight into how mathematics works, and the creativity and ingenuity required to structure proofs.

***Kurt Godel came along and constructed a mathematical sentence that could not be implied by the axioms. So now Wittgenstein’s statement is not only misleading, pejorative, lacking understanding and insight, it is actually rigorously wrong.

In a classic bit of irony, an alternative rigorous refutation can be constructed by a Turing Machine analysis. In theory, if the axioms imply the conclusion, there should be a sequence of statements starting with an axiom A and ending with a theorem T. The construction should go something like A > B1 > B2 >…> Bn > T (read A implies B1 implies B2 implies and so forth implies Bn implies T). These implications can be restated as an algorithm and a theoretical computer should be able to draw this conclusion. At first you might think so what? (At least I did). However it turns out that there is something called the Halting Theorem which ultimately allows one to test whether or not these algorithms, with given properties, do actually yield a result.

We know that this Turing Machine analysis provides an alternative proof for the Godel Theorems. In addition, from my readings anyway, it also shows that entire classes of mathematical statements can not be implied from given axioms.

The reason for the irony is that Turing was a student of Wittgenstein. Turing’s biography is also interesting but tragic.

Implications from a Kantian perspective would be that some Mathematics is analytic and some is synthetic.

  • Here I am using standard Boolean mathematics. (The same as Wittgenstein, Russell, or Hilbert used)

**I can not stress strongly enough that one should be extremely skeptical of anyone that misuses the standard meanings of a given word or phrase. (I understand the plus side, but my caution still stands)

“A” = “-A”

Few believe it; but it is true, and not, nonetheless.

Oh, my.

It’s only true if A = 0. Unless of course you are simply telling us that the symbol “A” is the same as some new “hybrid dash-A” symbol. Are you suggesting that in all cases “A” will ALWAYS equal “-A”, no matter what A is (this would be called stating an “identity”)? Or are you presenting a new declaration of what “A” will be called? In programming it is valid to have a line of code that looks like this:

x = 13; "this declares a variable and assigns it a value" print(x); "this displays the number 13 on the screen" x = -x; "this redefines the new value of x to be the opposite of what x used to be" print(x); "this displays -13"

This is perfectly valid code.


In terms of tautologies, i think you have the A-B relationships misinterpreted. Yes in a tautology if A implies B then B also implies A. But in that first argument that i quoted of yours, you have “x=5” being A, and “x>3” to be B. I’m pretty sure that in this case “A” is “x” and “B” is “5”. In this case A implies B always, and B implies A always.

“A” = “All”. (Which, by definition, must account for “Not All”)

Thank you for the good read Ed3. I’m still soaking it in. :slight_smile:

Hi CharlieGadfly,

Thanks for your response.

I have two comments. First, and maybe you know this, in standard Boolean mathematics you are not allowed to assume statements such as A = - A. You could try to prove your statement from axioms or other mathematical statements, but you can bet your bottom discretionary, disposable income that you will not prove such a non-Boolean statement.

However, outside that restriction I do have some sympathy for your statement. Somewhat analogously, I find the concept of THE Empty Set {} to be deeply troubling. I am in a tiny percentage of people that can not accept its existence and I am disappointed that Zermelo-Fraenkel Set Theory, usually abbreviated ZFC, axiomatizes its’ existence.

Hi Faust,

Thanks for your post.

Hi alexmabee,

In my declaration, if x = 5 then x is greater than 3, you are right to read A as “x = 5” and B as “x is greater than 3”. If we assume B, “x is greater than 3”, we can not conclude A, “x = 5”, because x = 4, x = 6, x = 7 et cetera are all counter examples.

Hi Bane,

Thanks for your comment. They are rare and I greatly appreciate it.

P.S. If you do end up with a question, please ask. I do screw up and I should be challenged.

First, I confess you are WAY over my head. As I have said here (or on some other thread) my opinion is intuitive. That said, let me re-articulate my position with some foundation:

It is my understand that a fundamental principle of logic is that an argument must go backward until a premise is found and agreed upon before the argument can then go forward in a search for truth.

Logic itself has gone back to one, fundamental underlying premise upon which it insists everyone must agree in order for logical argument to ensue. That premise is that you cannot have “A” and “Not A” in the same place at the same time.

Now, logic also provides that to avoid agreement, one need only attack a premise. I attack logic’s premise. And, since logic argues the burden of proof is upon the proponent of a position, then logic must prove that you cannot have “A” and “Not A” in the same place at the same time. It simply will not do for logic to demand that I prove otherwise.

It might be argued that I have already stipulated to an illogical position. However, this begs a question: Specifically, is logic defined by rules we give it, or does logic exist independent of what we would have it be? Think of Plato’s cave, shadows, reality, etc. Where upon that palate does logic lie?

I would argue that logic exists beyond the rules that WE give it, and that, at some future date, it may be logically proven that you can indeed have “A” and “Not A” in the same place at the same time. When that is done, it will be us that has changed, and not logic. We will have to confess a misunderstanding of logic that defies our out-dated and incorrect definition, and we will have advanced.

Indeed, it will be like the “laws” of physics that were always there, in spite of our failure understand or even agree with them.

I think that logic as we know it, like physics, breaks down at a certain point; specifically, pre-big bang and possibly in the center of a black hole. It’s not that logic and physics cease there, but that they are not something that we now know, but that we may know someday in the future. The only thing that is standing in our way is our steadfast refusal to think outside the boxes that we have created for ourselves, such as the unproven premise that you cannot have “A” and “Not A” in the same place at the same time.

I ask anyone who disagrees with this to please prove you can’t have “A” and “Not A” in the same place at the same time, and while they are doing so, please tell me what was, or was not, pre-big bang and/or in the center of a black hole.

Once they have done that, I would ask them to explain what it is about the nature of reality and/or the universe that proves that we are not, right now, pre-big bang or in the center of a black hole. And not. At the same time.

I, intuitively, believe in infinity and eternity. I cannot fathom how they could not be. And as such, I cannot fathom how they could fail to account for the absense of themselves, while still maintaining any kind of intellectual integrity. And that includes any “circular” (i.e. finite, non-linear) definitons of infinity.

I hate using words like “everything” because “thing” seems like a trap. I hate using “it” for the same reason. I have settled on “All” for lack of a better word. But, at the risk of someone trying to box me in with my limited vocabulary and inarticulation, I have to say, it’s all just too god damn big for anything to not be! And yes, I have to step back and admit that as such, I have to be wrong at the same time. It is “All.” It must be. And not. I think when we get over that, lots of worlds will open up to us.

And many people will be standing around slapping themselves in the forehead going: “It’s so simple! How did we not see that?”

“A” = “-A.”

Here’s why it’s a mistake:
(example only)
Premise 1: There is a system of symbols we’ll define as logic.
Premise 2: I exist (uses logic)
Premise 3: I will use the English language, which also relies on logic.
Premise 4: I wil type on my PC which uses logic.
Premise 5: I will type this claim, that is, logically I define it as a claim, and not “nothing”.

Claim: Logic is invalid, A and not A can exist at the same time.

Yes, your claim is a contradiction, you already accepted logic as a premise. You may proclaim you have not, or lie about it, or honestly not recognize it, but reality doesn’t beg to differ, it just is.

Logic doesn’t argue burdens of proof, it’s just a necessary condition for any symbolic system, and is frankly hardwired into the heads of animals of all kinds (which uses symbols). You ponder the origins of it, the origin of it is the universe, is the universe not a sufficient enough condition?

You mention Plato’s cave, if the greeks interest you, I would refer to Aristotles comments on those who logically attempt to refute logic. You’d have to say nothing, like a vegetable. I agree there are many silly philisophical claims, but logic is OK, even if I don’t admit it is :slight_smile:

-Mach

I’m going to have to think about that one for a while. :smiley:

Okay, it’s entirely possible I did not understand what you said, but, that said, here we go . . .

I don’t understand your five premises, especially preceded by a caveat that they are only examples. Please dumb them down a little bit for me. Fleshing them out, or firming them up as something other than an example might help me understand.

It seems Premise 1 stipulates to the notion that logic exists as we define it. That is a premise I did not accept when I ventured that logic exists beyond our contemporary perception of it. In other words, you have stipulated, with your premise, that logic is what we say it is and nothing more or less.

Premise 2 is not proven either. As to Premise 3, I can stipulate that you use the English language, but I’m not so sure about it’s reliance upon logic. As to Premise 4, I can stipulate that you use a PC and that the PC relies upon logic (I don’t know that it does, but I will assume all those ones and zeros are somehow logical).

Now, as to Premise 5, you say “I will type this claim, that is, logically I define it as a claim, and not “nothing”.” I guess this is where I lost you. Your claim is that logic is invalid, in that “A” = “-A.” However, that is only true if logic is as we define it, and if it does not exist beyond our definition (i.e. it’s in Plato’s cave).

If logic exists beyond our definition of it (as I argued above), then your premises could logically result in the claim: “Logic is valid, in that “A” = “-A”.

You can define it as you wish, but if logic itself is as you define it, and if it does not exist independent of your definition, then everything is whatever you say it is. That fits quite well within MY definition of “A” = “-A” but it seems to confirm my position, not refute it. It’s just another aspect of All.

Anyway, here is where I think you are mistaken:

You say I already accepted logic as a premise. But we have yet to agree on what logic is. You (I think, correct me if I am wrong) think that logic is what we say it is. I can accept that. But I can also accept that logic is something that we don’t yet know. In other words, I attack a premise that is part of a definition of logic that I don’t accept and which has not been proven.

Thus, my claim is not a contradiction since I have not accepted a premise used by your definition of logic. When you say: “reality doesn’t beg to differ, it just is” you get to the heart of the issue, and that is an assumption that reality just is. I think, therefore I am? I can stipulate to that, but it is a two valued orientation to then assume that reality then may also not be.

Is it your claim “Because reality is, it cannot then not be.” If that is the case, then I can see why logic must be as you define it, and not otherwise. Reality must be what we perceive it to be, in our cave, and not otherwise. Now, I know Plato was talking about “ideals” compared to imperfect copies, but I still think the metaphor is still instructive. Just because Plato (and the rest of us) can perceive of an ideal (and ideal logic, for example), that does not mean our perception of the ideal is accurate.

You say logic (as we perceive it, i.e. definitional) does not argue burdens of proof. I find that too convenient for logic (the ideal). Logic (definitional) can’t logically, on the one hand, require an agreement upon generic premises before some argument can go forward from a given point, and then on the other hand, claim exemption from the requirement when it comes to a premise upon which logic (definitional) relies. Let logic (definitional) prove it’s premise before we go forward with argument. In other words, definitional logic must match ideal logic or it is indeed invalid in itself.

You ask: “You ponder the origins of it, the origin of it is the universe, is the universe not a sufficient enough condition?” I’m not sure if you are using “you” rhetorically, or for me specifically, or if it even matters, and I’m not sure what the first “it” is that you refer to, but I will take it personally and say “No” it is not sufficient enough condition unless and until it accounts for the absence of itself.

Regarding Aristotles comments on those who logically attempt to refute logic, I’m afraid you’ll have to make his case for him if we are going to proceed.

When you say: “You’d have to say nothing, like a vegetable.” I don’t agree and I find no support for that in anything you have said.

When you say: “I agree there are many silly philisophical claims, but logic is OK, even if I don’t admit it is.” I can stipulate to that. In fact, I will go you one further and say that logic (the ideal) is OK even if we don’t know what it will be.

good read indeed…i don’t understand half of it, but it makes me wanna learn more! :slight_smile:

All I have to say is that there exists no matter but only perception.

Only because ‘=’ in programming doesn’t mean “left side is the same as right side”. It isn’t a logical statement, it’s a function that assigns the value on the right side to the left side. The only connection between the two meanings is that they use the same symbol.

A photon is a wave and a particle at the same time, synchronously.
So a photon is a wave and not a wave, a particle and not a particle, simultaneously.
It is all a matter of Perspective.

“All statements are true in some sense, false in some sense, meaningless in some sense, true and false in some sense, true and meaningless in some sense, false and meaningless in some sense, and true and false and meaningless in some sense.” -Robert Anton Wilson

Perspective.

I believe it is either the one or the other, not both at the same time. All the identity law is fake arguments fail because by their logic, there is no logic…but how did you get there?
Kudos for zeus here as well, one of the most common switching commands in in Fortran is something along the lines of A=x… x=B… which switches the place of A and B.

But it is so, nontheless. A photon is a wave, a particle, neither and all of the above… all synchronously. What you see depends on Perspective (who you are, how you see it, what you are looking for, the tools used to ‘see’)..
With all due respect for your ‘beliefs’, check it out for yourself. Quantum is the death knell for any ‘universal’ claim to the strictly local and pragmatic so called ‘laws’ of identity. They are refuted by quantum theory. The results of the double-slit experiment’s done it!
It’s a new game from here on.
Peace

Boolean is merely one type of logic, which in computing terms means piece of information (a bit) can be either a 1 or a 0, but not both. In this system, a proposition and its negation cannot both be true.

On the other hand, in the future we may see the birth of quantum computing, which would seem to make a case for dialetheism – a qubit, utilizing superposition and entanglement, is capable of representing a 1, 0, or both at the same time.

Then consider the possibility that the human brain is something of a quantum computer; what implications might this have on our understanding of truth and falsity? Or the role of perspective in their determination?

I believe you said simultaneously, meaning at the same time. I merely pointed out that that is incorrect.
You say that photons and quantum physics defy identity theory. I’m studying classical mechanics, and I personally find it peculiar for people to jump to conclusions based on a few fantastic studies about a fantastic field with no knowledge of the field that gave birth to it. Just vague statements about this and that. Anyway, the double slit experiment hasn’t defied identity laws, the photon was a wave at one point and a particle after the observation. Photon=photon. The fact that it can be both wave and particles is amazing (given the implication that time might not even exist!), but the law of identity stands in the same way that finding that the earth was not flat did not defy the identity law since the ‘true’ answer was ‘none of the above’. Until you say photon=/=photon, which somebody on this board attempted with half logic and no success, I can’t see how anything can defy its being.

Actually, that IS correct. Again, feel free to do the research. All the necessary data is on the net… IF you want to ‘hear it’. It does make chopmeat of your ‘god’, so I’d understand your inhibitions to do so. Otherwise, there is a whole new world, feel free to ‘catch up’.

Further, science is finding all ‘moments’ to be synchronous. ‘Linearity’ (naive realism) is a relic of Perspective. Hence the new ‘definition’ of the obsolete and clumsy notion of ‘cause and effect’ (beyond the ‘local’) is; “two features of the same event” rather than the linear notion of ‘cause and effect’.

I understand your Perspective. Quantum is the ‘critical update’ to ‘classical’ theory. There is no ‘empiricism’, no ‘objective observer’, etc… Classical without the ‘critical update’ is obsolete other than for a few local pragmatic uses, just like ‘cause and effect’. Quantum is the ‘critical update’ necessary for all other branches of science. Without it, they are already obsolete.

Just look around on the net. You will, if you are open minded enough to not be worshipping your particular ‘known’ practice and accept that it might be relatively ‘obsolete’. It is not my job to convince. Time and good understanding, and an open mind, will take care of that.
If not, it has well been stated and known that science progresses not by updating the 'known and attachments to ones field of endeavor, but death by death of the old and invested and attached (believers).
I’m not going to get involved with an argument. I also won’t get involved in a ‘flat earth’ argument, either, and that is how I see this.
Happy trails

Okay, let’s analyze what it is we’re debating here, namely the law of identity, particularly as it applies to photons (or elementary “particles” in general)

What is a photon? – A name we choose to refer to a particular phenomena we observe, apparently the fundamental constituent of light and radiation.

But upon what basis do we infer identity between any two photons? Or even one photon for that matter? Is a single photon observed really the same photon after having passed from one state to another, after making the probabilistic ‘jump’ in and out of time itself, from ‘point’ A to ‘point’ B?

What qualifications must be met here? It seems a certain level of understanding regarding the nature of the phenomena at hand is in order, no? Where are we with respect to that level? What must we understand in order to legitimately and with intellectual honesty make such claims?

Perhaps we’re overeager here to posit our laws of identity because they help us retain some of that headstrong confidence which your beloved classical mechanics bestowed upon man and his science?

Let’s be honest with ourselves: upon what basis do we conclude any two “things” (or events, or people, or…) are ontologically identical?

Perhaps, at bottom, two or more things can only humanly identical, or likewise humanly different?