Are numbers as real as rocks?

[Knotted]

But this is not the same as "greater" or "less"; or at least, it is not clear to me that is the case. I prefer to note that the idea of intensity of perception is unavoidable.

It seems to me that ordinality, or ordered sequence, is given to us by time; provided one can distinguish successive moments.

I may have slightly misunderstood what Jonti is saying. I'll just clarify that I was using 'well ordered distinction' to mean ordinality, not more general or continuous types of orderings. I try not to use words like 'ordinality' because it will put people off.

Cardinality, or magnitude, is given to us by sensory spaces. So the original intuition of mathematics (I'll call that "a distinction" for short) is supplemented by notions of sequencing or heaping together, precursors of fundamental operations.

I think this idea that we need this that and the other type of quality is analytic and so goes against (at least in spirit) Brouwer's synthetic notion of number. There is an almost mystical simplicity to what Brouwer is saying, and I'm not convinced of it.

 

[Jonti]

I feel Richard Feynman's "The Character of Physical Law", particularly the chapter on the great conservation principles, may be relevant to the question. Electric charge, and baryon number both come in units, and each is also conserved.

What this means in the case of electric charge is that the integer you get when you count all the positive charges in the universe, and all the negative charges in the universe, and then subtract the latter from the former never varies. It is just a counting proposition (as it is for baryon number).

As far as we can tell, the conservations of electric charge and of baryon number are both exactly accurate. The total never changes, not even by a single unit!

 

[Knotted]

I feel Richard Feynman's "The Character of Physical Law", particularly the chapter on the great conservation principles, may be relevant to the question. Electric charge, and baryon number both come in units, and each is also conserved.

What this means in the case of electric charge is that the integer you get when you count all the positive charges in the universe, and all the negative charges in the universe, and then subtract the latter from the former never varies. It is just a counting proposition (as it is for baryon number).

As far as we can tell, the conservations of electric charge and of baryon number are both exactly accurate. The total never changes, not even by a single unit!

I'm not sure what this particular example illustrates, though I think it does sound profound, but I think the broader question is when two quantities (or other abstraction) are mathematically equal is this just a semantic tautology or does the equation have real content?

 

[Jonti]

Well, the best equations are those that are almost tautological, but that turn out to have rich consequences. It's the same for the best scientific laws, like the theory of evolution. It sounds a tad tautological, but it isn't. It's massively useful when it comes to understanding the behaviour and development of ecosystems.

One of the consequences of variation with selection, according to Steve Jones (of "Almost Like a Whale" fame) is that it can create new information. I'm not sure we can usefully count that information in bits, but I suppose in principle that must be the case.

Anyway, apparently anything we can say, we can say in bits, so perhaps there's a sense in which any imaginable world must seem to have integer-style counting as its ultimate reality.

 

[118118]

cool, i wanted to talk about this again. can't concentrate at mo, but something about 1+1=3 not being true because true means 1+1=2.

that and i wanted to say that there's been alot of psychologism from knotted on this thread wrt 1+1=3.

i may try and wheel out the accepted arguments against this but can't concentrate at mo

 

[Knotted]

cool, i wanted to talk about this again. can't concentrate at mo, but something about 1+1=3 not being true because true means 1+1=2.

that and i wanted to say that there's been alot of psychologism from knotted on this thread wrt 1+1=3.

i may try and wheel out the accepted arguments against this but can't concentrate at mo

I don't think I ever put what I was trying to say particularly well. I'll try to summarise what my position was tommorrow morning. I've probably changed my opinion anyway.

 

[118118]

yeah cool mate.

 

[Knotted]

OK, let's start with the notion of truth. Not that I'm going to say what it is we mean by 'truth' (or by 'falsehood') but I am going to make a contention about how we use the notion of truth. And here it is:

Contention1: If a statement is said to be true it is conceivable that it may be false.

Justification: If it were possible to make a statement about something where it is not conceivable for it to be false then the statement would have no content as everyone would instinctively know it somehow. So contention1 is a corollary of:

Contention2: A true statement must have content.

Justification: If a statement has no content then it cannot have true content.

[cf Wittgenstein and the private language argument]

So now let's look at the statement:
1+1=2

By contention1, if we are to say it is true then it must be conceivable that it is false. Now there are two ways in which we can ask the question, "is 1+1=2 a true statement?" We can ask is it empirically true. We can ask is it logically derivable from axioms.

To deal with the first question, it is difficult to say what counts as empirical evidence. The question revolves around what we regard as a unit of something. We need to apply the notion of 'equality' in such a way that we can say "this unit equals that unit". When we say such a thing we are talking about distinct units, so there is an implicit notion of disctinctivness to our units. We can tell one unit from another. However there is a seemingly contradictory notion of sameness. We need to be able to substitute one unit for another without changeing the meaning of arithmetical statements. The 1's in 1+1=2 are not special 1's they are 1's in general - 1's in the abstract.

Let's suppose we are counting donkeys in a field. We know what a donkey is, don't we? So surely we can say, "I will group together all the donkeys in the field and count them." But what if there are also mules in the field, and perhaps a few hinnies with them. Would it be reasonable to count the mules as donkeys? That depends on what you mean by 'donkey', its rather up to you how you use the word 'donkey'. What if in the future donkeys have evolved into honkeys? When does donkey become a honkey? Is it meaningful to say 'as we can all agree on what a donkey is, then we can show that 1 donkey plus another donkey makes 2 donkeys'. No, it isn't. There is not even agreement amongst evolutionary biologists over what defines a species, therefore the evidence is spurious. Different people will count in different ways.

Now you could say that if there is agreement on the definition of the unit in question then we will all agree on the counting. Again this is not necessarily true in my opinion, although this is far more obscure. What if the act of counting affects what is being counted? This is possibly true in quantum mechanics (though that depends on your quantum mechanical interpretation). Is a particle in supposition with itself two particles or one particle? (More to the point is it even meaningful to talk about a particle at all?) Why is it not possible to count in a different (but consistent) way? As I have shown before, its perfectly possible.

Even then we can say that if we all agree on the interpretation then we can agree on the counting. But its at this point we realise we have ignored all the empirical 'evidence' and retreated to the plain subjective. If we all agree then we all agree. When we count we count like this, we all agree. This is what Wittgenstein would have called a grammar. We are using the grammar of arithmetic correctly. This is quite a different question to an empirical/ scientific question.

-----

Now I'm going to leave it there for the minute. This is turning out to be much longer than it is in my head, which is probably why I was never particularly clear in the first place. But anyway I've only summarised the easy bit. I've still got the logical-axiomatic evidence to discuss. Then there's more after that showing why I don't agree with the subjectivist ideas that I seemingly present.

PS Before anyone asks, I have been reading Philosophical Investigations in the interval.

 

[Jonti]

Even then we can say that if we all agree on the interpretation then we can agree on the counting. But its at this point we realise we have ignored all the empirical 'evidence' and retreated to the plain subjective.

We can only communicate the structure of our experience, not its phenomenological qualities. Seems to me this agreeing on the counting says that there's something inherently arithmetical about the structure of experience.

Or maybe that the necessity of using language to communicate, makes it appear to be that way (but to say so would be to assert that the structure of experience of a solitary consciousness would be qualitatively different somehow ... hmm).

 

[118118]

knotted, cheers, i myself am going to read wittgenstein hopefully soon. anyway, i can't comment much atm cos i can't concentrate to read your post.

next time, state your conclusion. followed then by your reasoning. it means that one does not need to apply as much concetration to get the gyst.

will get back to later.

 

[Knotted]

knotted, cheers, i myself am going to read wittgenstein hopefully soon. anyway, i can't comment much atm cos i can't concentrate to read your post.

next time, state your conclusion. followed then by your reasoning. it means that one does not need to apply as much concetration to get the gyst.

will get back to later.

That's sort of what I did first time round and it caused quite a lot of confusion. Part the problem is that I don't even know what I am going to conclude myself.

Anyway I haven't much time to say much for the next couple of days. But to summarise, I have attacked empirical mathematical realism, which is hardly a popular point of view anyway. What I have written so far is pretty straightforward. Feel free to talk about Husserl in the intermission. That would be interesting to me.

 

[Knotted]

We can only communicate the structure of our experience, not its phenomenological qualities. Seems to me this agreeing on the counting says that there's something inherently arithmetical about the structure of experience.

Yes. Good point, I'll continue where I left off at some point and I should deal with this point when I do.

Or maybe that the necessity of using language to communicate, makes it appear to be that way (but to say so would be to assert that the structure of experience of a solitary consciousness would be qualitatively different somehow ... hmm).

Maybe. I think language is structured quite differently to formal language including mathematics. Again I'm thinking of Wittgenstein here. There is no equivalent (as far as I'm aware) of family resemblance in mathematics. Linguistic thinking seems quite different, but sure there probably are overlapping logical/mental structures.

 

[118118]

(i did not mean to annoy)

 

[Knotted]

(i did not mean to annoy)

Its alright, I'm not annoyed.