On treeness of Oak1, and other things

[SpeedOfSound]

As can be seen with a google search for proof of the irrationality of √2, this is more or less the classical proof. Here is a link to the university of Utah offering a similar proof. Here is the definition of numbers from the same universtity site

None of this is contingent upon the definition and properties of a field?

[SpeedOfSound]

I'm so thrilled that you taught me about Tarski. He has apparently proven that you can't define a self-referential system. Therefore all math and logic is contingent on a meta-system.
The original meta-system is still cutting fish therefore empirical.

Tarski likes to say we have to assume or intuit or agree upon the meta definitions.

Do I have that right teach? Others?

[Little Idiot]

I will answer Graham, I am just doing a little checking first.
I was however making a joke about teaching you

I can save you some checking. Tarski is not your man. His concern over the formal definition of truth in logical systems led him to tighten things up. I believe one of his theorems was that you couldn't define a mathematical system inside of itself. This kind of bears out what CSG and SD were trying to tell you about math being a system where you get out what you put in.

I was not suggesting Tarski proved metaphysics was possible, I raised him as an expert writer who used the concept of non-contingent truth to show there was some weight behind this idea of truth which is independent of emperical observation. I extend from this point to say if there are known truths, not dependent on emperical observation but still known, they must be known by some other method; therefore there are other method(s) as well as empericism which can provide knowledge. None of this Tarski said (as far as I know) I am not suggesting he did say this, I am saying this, using him as a support for the starting point that there are non-contingent truths.

If we had some absolute truth lying around we never would have needed formal mathematical or logical systems. They are still arguing about some of this stuff. All of our words and systems have meaning tied up in our usage. Usage seems highly empirical to me.

That does not follow from absolute truth.
In metaphysics 101 I demonstrated that absolute truth is changeless, and therefore beyond time.
If, as I suggest (demonstrated) absolute truth is changeless and therefore beyond time, it may not have much to say about specific events within the realm of time, and there could still be (is) much need for science and maths within and applied to space-time.

I was just going to give you this math is a new way of knowing thing but you had to fuck around didn't you? To what purpose? So what if we can twist and turn until some of our ideas can be categorized into new ways of knowing. It still isn't metaphysics.

Unless of course you keep moving the goal posts on metaphysics as you and Jamest have been busy doing. We all hope and pray to X-God that you two will some day soon get on with your basis for your metaphysics.

I have no more interest in proving that there is a basis for metaphysics.
I am quite ready to do and discuss metaphysics instead, however.

My post early this morning was me throwing up my hands and saying "Fine, call me a fucking metaphysician too. Just get on with it."

I asked what problem you felt needed to be solved by your quest.

But answer GrahamH's question and let's spend a few months on this topic. I know you will have to have the last word as you always do. And it's always the wrong word.

I did respond to the post with the 'dilemma', that I presented to you, and I was waitng for your response.
But in any case, I think I have shown (again) a non-contingent truth. We now have two, (odd + even = odd for all whole real numbers) and (square root of two is irrational) we can add (pi) and others if need be.
I hope we dont need to spend months on it; its pretty silly to assert the only way we can get knowledge is by emperical method, to do so asserts there was no knowledge before emperical method for a start
I cant even remember why it was so important to show why there are other methods of knowing - can you remind me?

[Xamonas Chegwé]

simple proof for real numbers

Let n be any real whole number.
Then 2n is always even because 2n/2 = n which is a whole number. The definition of ‘even’ is met.
And 2n+1 is always odd because (2n+1)/2 = 2n/2 + ½ = n + 1/2 . The definition of ‘odd’ is met.

Therefore any odd plus any even is given by
(2n) + (2n + 1)
But since (2n + 2n + 1)/2 = (4n +1)/2 = 2n + ½ the definition of odd is met.

Therefore proving that for real numbers any odd plus any even will always be odd.

This does not prove that ANY odd + ANY even will be odd. It actually proves that any even number + the odd number that is one more than it, will be odd.

A correct proof is as follows.

All even numbers have the form 2z, where z is an element of Z, the set of integers.
All odd numbers have the form 2z + 1.

(Up to here, the reasoning was correct.)

BUT

Any even number + any odd number is given by: -

(2a) + (2b + 1), where a & b are any elements of Z. (Note the difference - in the original example, (2n) and (2n+1) where used - these are not ANY even and odd numbers but specifically consecutive numbers.)

To proceed:

(2a) + (2b + 1)
= (2a + 2b) + 1 (because addition is associative)
= 2(a + b) + 1 (because multiplication is distributive over addition)

Now, since the set of integers, Z, has closure under addition, (a + b) is also a member of Z and thus 2(a + b) + 1 is in the form 2z + 1, and is thus odd. QED.

NB. This proof applies to the set Z, which includes both positive and negative integers as well as zero. It is also valid for the set of Natural numbers, N, which contains only positive integers > 0. Parity (oddness or evenness) is not well defined for sets such as rational numbers, Q, real numbers, R, or complex numbers, C, so similar proofs cannot be constructed for these.

[SpeedOfSound]

I cant even remember why it was so important to show why there are other methods of knowing - can you remind me?

...None of this Tarski said (as far as I know) I am not suggesting he did say this, I am saying this, using him as a support for the starting point that there are non-contingent truths.

It was important for you to prove that metaphysics was possible. For some reason. I guess you had a feeling that if you showed one other way of knowing that CSG's OP would fall.

Now on Tarski. Unless you want to drop Tarski as your basis?

He proved just the opposite. Truth defined on a system must have the system defined on a superset of the system. That is a contingency.

[Little Idiot]

I'm so thrilled that you taught me about Tarski. He has apparently proven that you can't define a self-referential system. Therefore all math and logic is contingent on a meta-system.
The original meta-system is still cutting fish therefore empirical.

Tarski likes to say we have to assume or intuit or agree upon the meta definitions.

Do I have that right teach? Others?

Another way of understanding what he says is that if a single formal language can not define itself, or if a single formal languge can not define truth or a single formal language can not do (anything) then you just need two formal languages working in parallel to achieve said task.

I think its very far fetched to say any meta-language is emperical.

He certainally does not suggest self-referential systems dont exist.

[SpeedOfSound]

...

You a math guy? DO you want to check me on Tarski?

[Little Idiot]

simple proof for real numbers

Let n be any real whole number.
Then 2n is always even because 2n/2 = n which is a whole number. The definition of ‘even’ is met.
And 2n+1 is always odd because (2n+1)/2 = 2n/2 + ½ = n + 1/2 . The definition of ‘odd’ is met.

Therefore any odd plus any even is given by
(2n) + (2n + 1)
But since (2n + 2n + 1)/2 = (4n +1)/2 = 2n + ½ the definition of odd is met.

Therefore proving that for real numbers any odd plus any even will always be odd.

This does not prove that ANY odd + ANY even will be odd. It actually proves that any even number + the odd number that is one more than it, will be odd.

A correct proof is as follows.

All even numbers have the form 2z, where z is an element of Z, the set of integers.
All odd numbers have the form 2z + 1.

(Up to here, the reasoning was correct.)

BUT

Any even number + any odd number is given by: -

(2a) + (2b + 1), where a & b are any elements of Z. (Note the difference - in the original example, (2n) and (2n+1) where used - these are not ANY even and odd numbers but specifically consecutive numbers.)

To proceed:

(2a) + (2b + 1)
= (2a + 2b) + 1 (because addition is associative)
= 2(a + b) + 1 (because multiplication is distributive over addition)

Now, since the set of integers, Z, has closure under addition, (a + b) is also a member of Z and thus 2(a + b) + 1 is in the form 2z + 1, and is thus odd. QED.

NB. This proof applies to the set Z, which includes both positive and negative integers as well as zero. It is also valid for the set of Natural numbers, N, which contains only positive integers > 0. Parity (oddness or evenness) is not well defined for sets such as rational numbers, Q, real numbers, R, or complex numbers, C, so similar proofs cannot be constructed for these.

Thanks for that. Your proof is much more robust than mine, which I did say (in a panic of ego defensiveness ) it was a 'simple' proof.
I think I am right in saying mine is a simplified version of yours?
And even if mine is not a proof, but an error, your new proof actually proves my original point that an odd plus an even can be proven. Would you, then, agree that as I argue; any odd plus any even can be proven but not demonstrated emperically. Would you further agree that therefore there are other way(s) of knowing in addition to emperical method?

[Xamonas Chegwé]

...

You a math guy? DO you want to check me on Tarski?

Never heard of Tarski - philosophy is very much a mere dabblance for me - I don't know what you are all talking about in here a lot of the time (read: can't be bothered to wikipedia all of the references) but I do know bad maths when I see it!

I have a BEd. in Maths, am qualified to teach up to A level (anything pre-degree level) and my forté is Number Theory.

[GrahamH]

Can you give an example of a "non-contigent truth" and show it to be true?

I gave the example before of a non-contingent truth (although I didnt call it such) earlier, when I said emperical demonstration was impossible for 'an odd plus an even always gives an odd'
We had some fun with that; recall I suggested it cant be shown emperically but can be proved for real numbers as follows

simple proof for real numbers

Let n be any real whole number.
Then 2n is always even because 2n/2 = n which is a whole number. The definition of ‘even’ is met.
And 2n+1 is always odd because (2n+1)/2 = 2n/2 + ½ = n + 1/2 . The definition of ‘odd’ is met.

Therefore any odd plus any even is given by
(2n) + (2n + 1)
But since (2n + 2n + 1)/2 = (4n +1)/2 = 2n + ½ the definition of odd is met.

Therefore proving that for real numbers any odd plus any even will always be odd.

<snip>

All very good, but where did you get the numbers and arithmetic operations? All those proofs were contingent on a bunch of mathematical axioms, were they not?

Try something simpler. Prove that 1+1 = 2 is TRUE.

[Little Idiot]

There must be (at least one) other way(s) of accessing knowledge.

And we can see that for your money, it is the method of "ex recto assertion".

Emperical method can not access all knowledge which is known.

Look! Here comes one now!

Some knowledge is independent of emperical observation.

There's never just one ant at a picnic.

Non-contingent truths are independent of emperical observation.

But first you have to show how it is that they are true. If they are not based on axioms, on what are they based?

Do I? Or do I just have to establish that they are not based on empericism?
After all, that does away with your point "all we've got is emperical"
I am not, at this stage suggesting where they come from, only establishing at least one other method to emperical.

A statement of truth contains some knowledge, regardless of the definition of knowledge.

Yes, but first you have to show how it is true, otherwise you're just making an ex recto assertion.

Thats trivial to show; do you wish to argue against the point?

Why are we hearing about Tarski at this late date? Is a theory of truth contingent on doing research and name-dropping?

As I said, I refered to him as he used the truth I have based my agrumant on. This was not name dropping, but defence against your standard dismissal; 'thats out of your ass'.

Well some of us here are having a discussion, and as you will see SoS is quite happy to hear of Tarski. Why do you counter me with assertions that I am talking out of my ass, as opposed to actually dealing with my arguments? I may be unread and ignorant in comparison to your eminent self, however my limited but genuine attempts at forming arguments (be they flawed or otherwise) are far more significant than your bald face assertions (regardless of how correct or otherwise they may be) that I am speaking from my anus.

Your wit and intellect are both far superior to that required to assert such unsupported assertions, and I hope you will share both with us, in the form of constuctive arguments, be they points or counter-points to my own.

Note how much time and energy I have devoted to countering your point of some long time ago now 'all we've got is emperical' (WTE) while you think a single quip and mention of 'ex-recto assertion' is all thats required to dismiss my arguments or assertions.

[Xamonas Chegwé]

Thanks for that. Your proof is much more robust than mine, which I did say (in a panic of ego defensiveness ) it was a 'simple' proof.
I think I am right in saying mine is a simplified version of yours?

No. Your proof was wrong. It proved a specific group of cases but not the generality you claimed to be proving.

And even if mine is not a proof, but an error, your new proof actually proves my original point that an odd plus an even can be proven.

It can indeed be proven. It can be proven directly from the definition of an integer - and in many different ways.

Would you, then, agree that as I argue; any odd plus any even can be proven but not demonstrated emperically. Would you further agree that therefore there are other way(s) of knowing in addition to emperical method?

If I understand your use of empirical correctly (and I hope it is better understood by you than spelled) you are saying that the fact can be proven without having to try every combination of even and odd integers. This is indeed the case. This fact would be impossible to prove empirically, as there are an infinite number of integers.

Most proofs in mathematics are made using similar techniques and many cannot be made empirically. The ones that can be, tend to be counter-examples that prove hypotheses false.


However, pure mathematics is very much a special case (and number theory, into which the proof above fits, is a special case within that special case!) Mathematics is built entirely upon specific axioms and definitions and as such is internally consistent (even Gödel's incompleteness theorem does not contradict this.) Its application to the outside world is incidental (although incredibly useful) and as such, proofs in mathematics cannot be considered exemplary or typical of proofs elsewhere.

A proof in science means only that empirical results agree with predicted results within acceptable margins of experimental error. It is only ever a demonstration that the theory it is 'proving' is a reasonable model, not the truth.

In maths, a proof is far more than this. A mathematical proof, if it is sound, is absolute. It follows directly from clear definitions and previously proven lemmas. Take away the universe and maths will still be true.

So, yes, I would say that things can be known other than by empirical evidence but only very special and completely abstract things.

[GrahamH]

LI, if, as you suggest, mathematics is a route to metaphysical knowledge, do you think that String Theory, if it can be made fully consistent, will reveal the true nature of matter? Your current line of reasoning would seem to suggest you would agree, but your previous arguments suggest you would discount the idea.

Since its inception as the dual resonance model which described the strongly interacting hadrons as strings, the term string theory has changed to include any of a group of related superstring theories which unite them. One shared property of all these theories is the holographic principle. String theory itself comes in many different formulations, each one with a different mathematical structure, and each best describing different physical circumstances. But the principles shared by these approaches, their mutual logical consistency, and the fact that some of them easily include the standard model of particle physics, has led many of the world's greatest living physicists (such as Edward Witten, Juan Maldacena and Leonard Susskind) to believe that the string theory is a step towards the correct fundamental description of nature. In particular, string theory is the first candidate for the theory of everything (TOE), a way to describe the known fundamental forces (gravitational, electromagnetic, weak and strong interactions) and matter (quarks and leptons) in a mathematically complete system.

How might such a theory be shown to be wrong? Do you have another way?

Many prominent physicists, such as Richard Feynman and Sheldon Lee Glashow have criticized string theory for not providing any quantitative experimental predictions.

Is empiricism the only way to show that self-consistent mathematical theory is not a description of reality?

[Xamonas Chegwé]

Try something simpler. Prove that 1+1 = 2 is TRUE.

That is not simpler. That is asking for a formal definition of a number system (I presume either the natural numbers or integers, but not necessarily.)

To prove the statement above, it is first necessary to define both 1 and 2 unambiguously, as well as the process of addition and the meaning of equality. Such formal definitions do exist in number theory and if you are really interested, try reading Alfred North Whitehead & Bertrand Russell's Principia Mathematica. It might take you a few days...

[GrahamH]

Try something simpler. Prove that 1+1 = 2 is TRUE.

That is not simpler. That is asking for a formal definition of a number system (I presume either the natural numbers or integers, but not necessarily.)

To prove the statement above, it is first necessary to define both 1 and 2 unambiguously, as well as the process of addition and the meaning of equality. Such formal definitions do exist in number theory and if you are really interested, try reading Alfred North Whitehead & Bertrand Russell's Principia Mathematica. It might take you a few days...

Why is it not simpler? Do you mean it is a more difficult task? Aren't the other proofs (or attempts at proofs) contingent on these basics? If they are, and remembering that LI is after non-contingent truth, then a non-contingent proof of number theory is a necessary part of his other proofs.

Will Number Theory reveal something about the nature of reality?