On treeness of Oak1, and other things

[GrahamH]

On a related topic, are the law of identity and law of non-contradiction provable non-contingent truths about reality?

[Little Idiot]

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.

Fair cop.

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)

<cough>

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.

Thats exactly what I said when I first brought it up, only by an infinite numbver of tries could it be demonstrated emperically.

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.

And I would say this is a fairly typical view of the mathematicians.
Maths functions anywhere its axioms hold, regardless of the emperical world.
Maths deals routinely with abstract things - a point some physicalists seek to deny.

Obviously I am happy that you are proving my original point about odds and evens, (even if you squish my 'proof') and I am happy that someone is finally agreeing with me that things can be known other than by empirical method.

Now if only I can recall why we were arguing about that....

[Xamonas Chegwé]

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?

Mathematics is only as good as its initial assumptions. In the case of the simple number theory examples given previously, the assumptions are formal definitions of number and postulates such as "two things that are equal to the same thing, are equal to each other."

In the case of string theory, assumptions about the nature of reality itself have been made and the maths bolted on to it - this is the case in most applied mathematics. Newton's equations of motion were based on assumptions made from extrapolation of empirical data. These equations were later proved to be flawed when Einstein's theories of relativity predicted aberrations in the orbit of Mercury and the bending of light from stars as they passed behind a solar eclipse.

Newton's maths was not at fault. The underlying assumptions were.

String theory might turn out to be bollocks but the maths based upon it is sound because it is consistent providing that the assumptions it is based upon are true. The problem is that there is nothing comparable with Einstein's predictions that will demonstrate the truth of those assumptions, not yet in any case.

[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?

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.

Allrighty then. So we spend some more time on this.

[GrahamH]

<snip>
Maths functions anywhere its axioms hold, regardless of the emperical world.
Maths deals routinely with abstract things - a point some physicalists seek to deny.

Obviously I am happy that you are proving my original point about odds and evens, (even if you squish my 'proof') and I am happy that someone is finally agreeing with me that things can be known other than by empirical method.

Now if only I can recall why we were arguing about that....

How do we know where the axioms hold? How do we tie the abstract to the real?

[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...

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?

It is a far more difficult task. But it has been done. Asking LI to prove number theory is equivalent to Carl Sagan's apple pie from scratch. The universe exists and there are already apples in it - creating another one is redundant.

LI's use of number theoretic proofs as an example do not prove his later points - I already pointed that out in another post. Propositions in number theory can be proven absolutely because the definitions of number are precise and the mathematics based upon it is internally consistent. This is the exception rather than the rule. In applied maths, the definitions are based on empirical measurements and assumptions not on precise definitions.

Number theory reveals things about number theory - nothing else.

[SpeedOfSound]

And I would say this is a fairly typical view of the mathematicians.
Maths functions anywhere its axioms hold, regardless of the emperical world.
Maths deals routinely with abstract things - a point some physicalists seek to deny.

Obviously I am happy that you are proving my original point about odds and evens, (even if you squish my 'proof') and I am happy that someone is finally agreeing with me that things can be known other than by empirical method.

Now if only I can recall why we were arguing about that....

I don't know how and the fuck I got around to making arguments about empirical data so I am going to have to correct that. I don't actually know what the fuck sense we are using that word in.


But more later about a revision of my claim. It wont be a hell of a lot different.

[Xamonas Chegwé]

<snip>
Maths functions anywhere its axioms hold, regardless of the emperical world.
Maths deals routinely with abstract things - a point some physicalists seek to deny.

Obviously I am happy that you are proving my original point about odds and evens, (even if you squish my 'proof') and I am happy that someone is finally agreeing with me that things can be known other than by empirical method.

Now if only I can recall why we were arguing about that....

How do we know where the axioms hold? How do we tie the abstract to the real?

This is the crux. In pure mathematics, such as number theory, geometry, calculus, etc. the axioms are abstract. Applied mathematics attempts (often incredibly successfully) to model observed effects in the real world using mathematics but, as I keep reiterating, it is only as accurate as its initial assumptions.

[Xamonas Chegwé]

Right. Enough for me. I only popped in here to correct someone's maths homework!

It's been fun but you guys will eat up all of my evening if I am not careful.

Enjoy the rest of your debate.

[GrahamH]

Right. Enough for me. I only popped in here to correct someone's maths homework!

It's been fun but you guys will eat up all of my evening if I am not careful.

Enjoy the rest of your debate.

Thanks for the informed contributions.

[Little Idiot]

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?

Actually Graham, with respect, I got what I want, what I was after.
I only brought in Tarski and non-contingent truths to prove the point I have been arguing against for (seems like) weeks which was made by SD amongst others and defended by (seems like) virtually everyone summed up in the point "all weve got is empiricism"

[Surendra Darathy]

I only brought in Tarski and non-contingent truths to prove the point I have been arguing against for (seems like) weeks which was made by SD amongst others and defended by (seems like) virtually everyone summed up in the point "all weve got is empiricism"

Well, then, all we've got are empirical observations and axiomatic systems. Oh, and woo. AKA, ex recto assertions.

Set-Theoretic Version: Some of it, plus the rest of it, equals all of it.

[Little Idiot]

I only brought in Tarski and non-contingent truths to prove the point I have been arguing against for (seems like) weeks which was made by SD amongst others and defended by (seems like) virtually everyone summed up in the point "all weve got is empiricism"

Well, then, all we've got are empirical observations and axiomatic systems. Oh, and woo. AKA, ex recto assertions.

Set-Theoretic Version: Some of it, plus the rest of it, equals all of it.

But for this evening, at least, I bask in the comfortable glow of being right.
Although you may or may not choose to give me the tid-bit of actually conceeding the point, I have shown that there is at least one other way of aquiring knowledge, and refuted the statement that the only way of aquiring knowledge is by emperical method.

This is not to suggest that knowledge of the physical world need not normally be in alignment with empirical evidence, nor to suggest empirical evidence is any less than highly important when dealing with the physical environment. This is, however, a disaster for the anti-metaphysics position, as it proves a statement can be true, and can provide knowledge even without empirical evidence to support it. A metaphysical statement about the non-physical does not need the suport of empirical evidence, and not having such evidence can not be used to demonstrate the statement is untrue, nor that it is an error.

Even the very term 'woo' has been shaken to its foundations; that without empirical evidence, which was previously disregarded as woo - because of the eroneous belief that without empirical evidence equals false - is no longer automatically dismissed, because lack of empirical evidence no longer equates with false when dealing with some areas of knowledge.

[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?

Actually Graham, with respect, I got what I want, what I was after.
I only brought in Tarski and non-contingent truths to prove the point I have been arguing against for (seems like) weeks which was made by SD amongst others and defended by (seems like) virtually everyone summed up in the point "all weve got is empiricism"

You forgot fairytales. We have always had those too. Fairytales can be internally consistent. They can 'make sense', but most of us don't think they are a 'different way of knowing' that reveals 'truth about reality'. The Brothers Grimm were not doing metaphysics.

Your task is to show that abstract formal language with internally consistent rules says anything about 'reality'.

So let me restate a few questions you seem reluctant to answer.

1. Do you think that the mathematics of String Theory could, in principle, reveal the absolute truth of what matter is?

2. Can you suggest any means of connecting abstract ideas with 'reality', other than empiricism?

3. Do you realise that mathematical proofs are contingent on mathematical axioms which are either derived from observation, or simply invented?

[Little Idiot]

Actually Graham, with respect, I got what I want, what I was after.
I only brought in Tarski and non-contingent truths to prove the point I have been arguing against for (seems like) weeks which was made by SD amongst others and defended by (seems like) virtually everyone summed up in the point "all weve got is empiricism"

You forgot fairytales. We have always had those too. Fairytales can be internally consistent. They can 'make sense', but most of us don't think they are a 'different way of knowing' that reveals 'truth about reality'. The Brothers Grimm were not doing metaphysics.

Your task is to show that abstract formal language with internally consistent rules says anything about 'reality'.

Oh no Graham, you dont distract me that easily.
My point is made; There is at least one other way of aquiring real knowledge without empirical evidence. You are no longer able to say that a metaphysical statement is untrue because if lacks empirical evidence. You are no longer able to say absract reasoning is automatically in error because it does not have empirical evidence. Thats what I have got.
You keep Brothers Grimm, thank you.

I care not a hoot if 'abstract formal language with internally consistent rules says anything about 'reality' or not'; its my position that they can not do so, because of what reality means to me - as I showed in metaphysics 101 'absolute truth' is changless and therefore timeless, I use the Vedantic definition of 'absolute' reality as that beyond time. Absolute truth describes timeless reality.
This, to me, is distinctly different to existent which I think is what you mean by the word reality in your sentence I quote above.
Maybe the sentence should read 'abstract formal language with internally consistent rules says anything about 'existence' or not.' In either case, I dont think its my issue to try and establish that.

So let me restate a few questions you seem reluctant to answer.

1. Do you think that the mathematics of String Theory could, in principle, reveal the absolute truth of what matter is?

2. Can you suggest any means of connecting abstract ideas with 'reality', other than empiricism?

3. Do you realise that mathematical proofs are contingent on mathematical axioms which are either derived from observation, or simply invented?

I am reluctant to answer these questions for several reasons, not least of which is the use of absolute truth and reality in ways I am not able to agree to. But if pressed I will offer tentative responses, bearing in mind my objection to the terms of the questions.

1. Do you think that the mathematics of String Theory could, in principle, reveal the absolute truth of what matter is?
First reaction; How the fuck do I know? I am just a science teacher.

There is no absolute truth to matter, matter does not exist.
Do I think it can reveal fundamental information about the physical world? Absolutly, it is far too powerful to be total BS. Is it the 'ultimate theory of everything' it may well be a big step on the way.

2. Can you suggest any means of connecting abstract ideas with 'reality', other than empiricism?
Emiricism is not concerned with reality, it is concerned with existence, so no there are no ways that empiricicm can 'connect' anything with reality. The question is flawed and meaningless.
'existence', not 'reality' is to do with empiricism.
I have no problem with empiricism when used to work in the physical realm.
I see no way empiricism can possibly make an attempt to 'connect' with the timeless reality.
If we are to deal with timeless reality, I think the question should be reversed; is there any way we can 'connect' with reality other than through abstract ideas?

3. Do you realise that mathematical proofs are contingent on mathematical axioms which are either derived from observation, or simply invented?

I realise many mathematical proofs are based on axioms derived from observation, others are derived from axioms not based on observation.