h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

oxymorons be daed!

Whaddaya mean, being a witch doesn't sound normal?

Or did you mean the stuff about burning?

"If something exists in the Real World, you *must* be able to say where it is."

Not true. Abstractions can have objective reality. An abstraction, by definition, has no physical reality, but it can still have objective reality if it is independent of the context from which you view it. Mathematics is one such abstraction. I'm not sure how many others there might be.

"I don't think you can say that mathematics is objective and independent of human activity. Axiomatic mathematics is about what you could "in principle" write down and agree upon. It is a set of rules that all mathematicians have to adhere to. It is a subjective matter whether two mathematicians agree with a proof or not, although in practice there is general agreement among mathematicians."

Apart from when flaws are found in proofs, all of mathematics should be considered true by a mathematician worthy of the name. The whole point about mathematical proof is that it is definite beyond gainsaying, unlike statistical or physical proofs. Unless a mathematician can find a flaw in a proof, they've got no business saying it's not true.

"I'm saying that mathematics is defined in terms of human relationships and cannot then be said to be independent of human activity. The whole point about axioms is that they provide a final arbiter in an argument between two mathematicians (at least in principle). Without the humans and their argument, axiomatic mathematics is meaningless."

This is only true if define maths to be the process of argument between mathematicians. I wouldn't use that definition. I would define it as a coherent and internally consistent logical structure.

"Secondly, even ignoring that problem, there is nothing that connects the mathematical "two" to empirical reality."

This is a problem how? Mathematics is an abstraction. I don't see why it needs to have physical reality to be considered objective. In fact, I would count physical reality as a point against objectivity as I am unconvinced by the concept of an objective physical reality. As in phenomenology, the concept is valid wherever it comes from.

"Most modern mathematics uses the Zermelo-Fraenkel axiom set which consists of 9 (or 10 depending on whether you include the axiom of choice) axioms, one of which is the existence of the empty set."

Actually, what these axioms are a very exacting definition of sets, element of sets and how they behave. An assumption in mathematics is a fact you take to be true at the start of your proof. If your assumption is true, then your proof will hold. If it isn't, your proof fails. These axioms are not assumptions; they are definitions. The first (and only) assumption we are left with is the existence of the empty set. You can try and build a structure with different axioms, but you will find yourself forced into a contradiction or back towards the axioms.

"If you do look at that site you'll see that the axioms are about what can be written, they are about symbols."

No! The symbols are just the language we can use to describe the concepts. There can be other languages to say the same thing. The point is, any life-form considering mathematics will arrive at the same mathematics that we have. There's no helping it. The way they arrive at it may be different, the language they use will be different, and the areas they explore will be different, but the underlying code really is independent of the mind that considers it.

I think where we're butting heads here is the definition of independent.

When I say 'maths is independent of the human mind' I mean:

'There is no qualitative link between maths and the human mind. If you change the human mind, or replace it with another mind, or with no mind, mathematics remains the same'

It's the statistical definition (and commonly used, I think).

What you're driving at (If I've got it right) is that mathematics is not autonomous and cannot 'exist' without consciousness. However, as maths is an abstraction it makes no sense to try and link it to the phenomenal universe in this way.

Mathematics is fundamental because, it turns out, it is the only such structure that is internally consistent (This is partly because anything that is internally consistent becomes mathematics). It doesn't matter where our thought processes start out, we are forced into mathematics because it is the only option that is coherent. Similarly, the phenomenal universe seems to follow mathematics because it is the only option available.

In summary:
Abstractions can have objective reality.
Broad-ended definitions can have meaning and exist before a mind considers them. (i.e. atheism can, but non-Catholic can't because it is specific).
Therefore, anything that does not acknowledge a deity is atheistic.
A chair is atheistic. And, to bring the argument full circle, primitive man was atheist.

Whew. How on earth did we get side-tracked into the nature of mathematics?

It's an old subjectivity/objectivity argument.

Yours is a pretty good response, though I'm sure PC will still find something wrong since however you cook it, I think there is still at least one assumption in there.

AIUI, an abstraction is a mental phenomenon, necessitating a mind within which it may take place. If something exists in the Real World, you are able to say where it is.

A rock and an X-ray both exist in the Real World, and can be located therein.

An idea can *apply* to the Real World, maybe even usefully and accurately, but it has no *existence* outside of a mind.



I'm sorry, but I can't make sense of that.

Pattern-chaser

"Who cares, wins"

"AIUI, an abstraction is a mental phenomenon, necessitating a mind within which it may take place."

Not really. It's a concept divorced from physical reality. At least as I think it.

"

I'm sorry, but I can't make sense of that. "

If it only has existence within minds, then it must be subjective, as for all such things there can be disagreement between minds about them. If something is objective then two minds, no matter how disparate, will agree on it. Therefore, if it is objective then it must (in some sense) exist outside of any minds.

So, if your abstraction is objective (i.e. unaffected by the nature of the abstracting mind) it must have some kind of objective reality.

There's only a very small class of abstractions that meet these cirteria, though.

In some ways, these objective abstractions are more 'real' than the world we experience.

, I missed the change in terminology. I have made no claims regarding "objective reality" (whatever that may be! ).

I have said that an abstraction - such as mathematics, or more simply, "two" - has no existence in the Real World. It exists only inside one or more human minds. The Real World contains X-rays and rocks, but it contains no abstractions at all.

I agree with you that abstractions, by this reasoning, must be subjective.

<[an abstraction is] a concept divorced from physical reality.>

Exactly: it has no existence in the Real World.



Yes...





<...if it is objective then it must (in some sense) exist outside of any minds.>

Yes. Therefore an abstraction cannot be objective. QED.

Pattern-chaser

"Who cares, wins"

"Mathematics is fundamental because, it turns out, it is the only such structure that is internally consistent (This is partly because anything that is internally consistent becomes mathematics)."

Actually, it's not known whether mathematics is consistent or not. You can prove consistency of some axiom sets using different axiom sets (for instance, it's apparently possible to prove the consistency of arithmetic using transfinite induction), but an axiom set cannot prove its own consistency. In fact, it is quite possible that we will discover an inconsistency at some point in the future, although most mathematicians believe that we won't.

"You can try and build a structure with different axioms, but you will find yourself forced into a contradiction or back towards the axioms."

Even given the consistency of our axiom sets, there are problems. For example, there is the famous example of the axiom of choice. It's quite an obscure axiom, but it has been proved that IF it is consistent to hold that the axiom is true, it is also consistent to hold that the axiom is false. This undermines the concept that there is a single thing called mathematics. In fact, there are many different mathematics depending on the axiom sets we use.

"The first (and only) assumption we are left with is the existence of the empty set."

Simply not true. The axiom of infinity asserts the existence of the set of natural numbers. It is possible to prove the existence of a set of the first n natural numbers (for any n) without using the axiom of infinity, but it is not possible to assert the existence of a set containing all of them without it.

"What you're driving at (If I've got it right) is that mathematics is not autonomous and cannot 'exist' without consciousness. However, as maths is an abstraction it makes no sense to try and link it to the phenomenal universe in this way."

What I am saying is that logic doesn't exist without people, it is an invented concept, albeit one which has been extremely useful (to people). The universe itself has no need for logic, and if there were no people in it there would be no logic. Logic, and mathematics, are a phenomenon of human thought. To assert anything about logic is therefore to assert something about human thought and human relationships. You can assert that any alien race would have the same logic and mathematics that we do, but it is only an assertion. Does a dog have the same logic and mathematics as us?

"Whew. How on earth did we get side-tracked into the nature of mathematics?"

Well, I think you were making rather grandiose claims about mathematics ("I say again: out thought processes are a product of mathematics, not vice versa.") which I couldn't let go unchallenged . Mathematicians have a tendency to make wild assertions about mathematics which go unchallenged because few people feel informed enough to contradict them. I'm acutely aware of this phenomena because I used to do it myself, but there really is no excuse for it (in my opinion).

One of the things I always surmised about maths is that it's the only thing we can know to be true, because it's not based on observations of the 'real' world. Because it doesn't have to conform to these wild and erratic things, it can be true and consistent quite easily.

It seems quite fortunate that we can model what we perceive of many real-world phenomena with maths.

In Mathematics consistency IS the truth. And it's not easy. That's why so many of us (and I'm one of the "us") find it difficult. But from whence comes this demand for consistency? And from whence do we get the idea of inviolable rules? The universe cannot be incontrovertibly shown to have either rules or consistency; these are regulatory principles of human understanding - which is not to say that they have not proved useful, of course, or that we could use, or even have, other guides, and we view the universe through these particular lenses.

Noggin

"I agree with you that abstractions, by this reasoning, must be subjective."

No! Not all of them. A subjective statement is linked to a certain viewpoint. And objective statement isn't. Contradictory subjective statements can both be 'true', contradictory objective statements can't both be true.

So, if an abstraction meets this objectivity criterion, then there must be some objective truth independent of any mind. So, there is some 'reality' underlying this abstraction, even if it has no physical presence.

The point I was making is that abstractions are not necessarily subjective.

"It seems quite fortunate that we can model what we perceive of many real-world phenomena with maths."

That's what I was getting at in one post. If our thought finds itself inexorably drawn into mathematics, then it seems likely that the real-world is also inexorably drawn to the same processes.

"In fact, there are many different mathematics depending on the axiom sets we use."

BUT, our axioms are simply definitions we find it convenient to use. If we build a new structure using different axioms is is not a different mathematic, but another branch of the same mathematics.

"The axiom of infinity asserts the existence of the set of natural numbers."

I understood that von Neuman's Brainwave showed that you can prove the existence of the natural numbers (and the arithmetic upon them) only assuming the existence of the empty set.

"What I am saying is that logic doesn't exist without people, it is an invented concept, albeit one which has been extremely useful (to people). The universe itself has no need for logic, and if there were no people in it there would be no logic."

I disagree. The universe's behaviour (as far as we can tell) is governed entirely by logic, mathematics and statistics. It has no 'need' for it (whatever that means), but it is controlled by it.

There's basically two ways to look at it:

* There is something truly fundamental about mathematics which means the universe and thought are compelled to obey it (such as it being the only consistent system)

* Our mathematics is only one of many possible ones and we based it on the observed universe.

However, as mathematics encompasses all such consistent systems, then I find it difficult to subscribe to the second.

The big question in cosmology of why the physical laws are as they are has its most compelling (IMO) solution in 'because it's the only way they can be'.

"Mathematicians have a tendency to make wild assertions about mathematics which go unchallenged because few people feel informed enough to contradict them."

Possibly. I still think mathematics is fundamental. If there were any alternative axioms that led to a consistent system I would have thought they would have been published by now. It's pathological enough to interest mathematicians, after all.

"which is not to say that they have not proved useful, of course, or that we could use, or even have, other guides, and we view the universe through these particular lenses."

Not in all cases. You can't change the value of pi without doing something really twisted to mathematics. It doesn't matter what lense you use, in this universe you're going to get the same vlaue of pi. Similarly, elementary logic would still be elementary logic. There may be different languages for describing it, there may even be different ways of approaching it, but ultimately they'd be different parameterisations of the same underlying concept.

OK, let's consider a hypothetical abstraction which is (somehow) objective. If it's objective, it does not depend on human minds. Therefore it must exist somewhere, somehow, outside of human minds. Where is this 'somewhere'?

Pattern-chaser

"Who cares, wins"

"Therefore it must exist somewhere, somehow, outside of human minds."

Exist, yes. But not physical existence as abstractions (by definition) have no physical presence.

A 'universal truth', an objective fact (rather than object) is both abstract and non-physical. Remember we're objective in the objective/subjective sense rather than the 'like an object' sense.

I don't see what's so difficult about this.

Does wind exist? Yes. But it has no physical presence. We can only identify it by its effect on air, which has physical presence.

The universe acts as it acts. It is not "compelled" to obey anything. Mathematics is a human invention, which helps us to understand and predict how the universe acts. If there's a contradiction here, it's mathematics which is wrong, and must change, not the universe. The universe, by definition, is correct.

Pattern-chaser

"Who cares, wins"

I thought you'd do this. You've commented on an incidental statement I made, and ignored the (unanswerable! ) question I asked at the end of my paragraph:

PC:

Where is this 'somewhere'?

Pattern-chaser

"Who cares, wins"