Are numbers as real as rocks?

[Knotted]

I have only read a secondary text on the Husserl's philosophy of arithmetic, but he abondoned it so I wasn't looking for all the answers. But that is where most of my feelings on the subject come from.

I’ll come back to some of the points raised in some of the above posts. At the minute I’m reading up on Husserl. There is a great irony here. I’ve read Frege but not Husserl, and I think 118118 has read Husserl but not Frege. Yet in our disputes I take the side of Husserl and 118118 takes the side of Frege!

A lot of the things I have been saying are related to the use of the notion of identity in mathematics. Two things are identical if and only if you can substitute one for the other. This is certainly what happens in mathematics, but is there a deep meta-mathematical reason why this should be?

When I talked about a type of arithmetic where no two units are the same, I was thinking specifically of violating Frege’s belief that you can substitute any unit for another. In the arithmetic I was talking about you would have to keep track of the different units. So 1+1’ and 1+1’’ would have to be labeled by two different ‘2’ symbols. There is nothing to be said in favour of this new arithmetic in terms of practical mathematics - the statements in it would be horrendously baroque - but in terms of mathematical ontology I can’t see why it is any less valid than the usual arithmetic.

This essay is of interest:
http://perso.orange.fr/rancho.pancho/Sub.htm
In particular:

“In another argument, Husserl alludes to the problems that arise when one begins examining the grounds for determining the equality of two objects (pp. 108-09). One can declare two simple, unanalyzed objects equal without much further ado, he notes. But there is a certain ambiguity in ordinary language with regard to complex objects. If two objects are the same, then it follows that they must have all their properties in common. But the inverse does not seem to hold. Sometimes two objects have their properties in common and we still do not say that they are the same.”

Also my reading of Stanford Encyclopaedia of Philosophy on Husserl indicates that Husserl would not have had any problems with true statements about fictional entities (even including Pegasus!):

“Even objectless (i.e., empty) intentional experiences like your thought of the winged horse Pegasus have content. On Husserl's view, that thought simply lacks a corresponding object; the intentional act is merely “as if of” an object.”
http://perso.orange.fr/rancho.pancho/Sub.htm

Again I think I'm in agreement with Husserl here and 118118 is in disagreement.

However I’ve got to say that I don’t really understand phenomenology and ‘intentional experiences’. I’m immediately sceptical about such things as ‘units of consciousness’. Why should consciousness come in units?

 

[Knotted]

I think the suggestion is that the (perception of the) passage of time implies two-ness because this is now, and that was then. But, to my mind, it's the perception of difference that's really fundamental. It is only by making a distinction that mathematics become possible; and that implies a cleaving of a previously unified space.

The first distinction, one might say, creates duality.

As I understand it Brouwer thought that time is synthetic a priori but space
is not. I'm not sure why he thought this, but it seems to be why he emphasised temporal difference rather than spacial difference.

Incidently this line of thinking seems to be related to some of the things in the Revising Reality thread. Our brodix and Lee Smolin seem to be getting all Kantian about time. Maybe Brouwer is another philosophical kindred spirit.

 

[Knotted]

To fly off at a tangent again. Let’s think about old fashioned shoot-em-up computer games where the score goes up in multiples of 1000. It’s a pointless attempt to make the game seem more exciting, the score could be calculated by ignoring the last three 0’s and for all practical purposes this would be the same. So we can enumerate things in terms of 1000 just as well as we can enumerate things in terms of 1000’s as we can in terms of units.

But units have a property that 1000’s do not. We can enumerate the enumeration with units. We can say 1x56=56 but we cannot say 1000x56000=56000. We have the identity 1xA=A but we do not have the identity 1000xA=A.

Therefore arithmetic is not merely about enumeration it is also about the enumeration of the enumeration. However, this second level of enumeration does not have the same ideal qualities as the first. It breaks down when 0’s are enumerated.

The above does not answer the Jonti’s original question but I hope it gives an insight into what the question was asking in the first place.

 

[Knotted]

Doesn't that answer the question as to whether numbers are real though? To state what the subject matter of mathehatics is, is to say whether they are eternal or not, mind independent or not, empirical or not.

I think the question, "what is the subject matter of mathematics?" is very difficult in its own right. I haven't answered it, but I hope I've given some insights to what the answer should look like. But I agree that the question of realism would be answered with minimal effort when the subject matter question is answered.

 

[118118]

I think the question, "what is the subject matter of mathematics?" is very difficult in its own right. I haven't answered it, but I hope I've given some insights to what the answer should look like. But I agree that the question of realism would be answered with minimal effort when the subject matter question is answered.

So when you say subject matter of mathematics, you mean what mathematicans do, not numbers

 

[118118]

But units have a property that 1000’s do not. We can enumerate the enumeration with units. We can say 1x56=56 but we cannot say 1000x56000=56000

I'm not sure what you mean by enumerate enumerations. Its easier to tak in tens. I can count to 500 in tens, or I can count to 500 in ones. I can then count how many times I counted, in tens, or ones. It would seem that we can enumerate enumerations with both ones and tens. I don't see what you do mean by 'enumerate the enumeration'. It seems like a way to crowbar in an alternative mathematics by using simple mathematical terms like 'enumeration' to mean some completely different thing/act.

 

[Knotted]

I've googled and skimmed a few bits of text on thin realism. Its "thin" because numbers do not have any properties not ascribed to them by mathematicians. E.g. they are acausal, do not have mass, etc. but I don't see how this is any different to full realism: what properties do full realist ascribe that thin realists do not.

I think that the thiness of the realism refers to the belief that the properties of mathematical objects cannot be derived from physical or metaphysical properties of physical or metaphysical objects. Mathematical objects are the way they are because they are defined to be that way.

Take for example the axiom of uniformity in set theory. It states that no set may contain itself. Considering that having sets of sets is allowable this 'axiom' seems to my mind quite artificial from a meta-mathematics point of view. However from a practical mathematical point of view it avoids paradoxes and it does not restrict the subject matter to any great deal. I think it is there for practical reasons rather than ontological reasons.

So for a thin realist the practical reason is all that matters.

But a robust realist would have to agonise over whether the axiom of uniformity is "really true".

I wasn't 100% satisfied with the idea that they have no explanation of how we acquire mathematical truths other than the working of mathematics. The explanation just terminates there, and I'm not used to explanations stopping like that (I find it unsatisfactory in science, tbh)

Yes it is unsatisfactory. I'm not satisfied either. I would see that as an indication that there is still a great deal of work to be done rather than that the work is going in the wrong direction. I would expect there to be a great deal of work to be done anyway, in fact I strongly suspect that the work is infinite. We'll neve quite be able to pin down exactly what we're talking about in order to answer clear questions in a clear manner.

Am I right that thin realists only differ from realists in athat they do not ascribe to realism in truth, but only realism in ontology?

No. I'm not sure how easy it is to generalise, but I would see a thin realist as ascribing 'a little bit' to realism in truth and 'not really' to realism in ontology and 'somewhat' to realism in existence.

I admit to be a bit confused that they often compare maths to astrology.

I would have thought you would like that bit. Its an argument against irrealism.

I just don't see science working if the same entities were being referred to and yet 1+1=0. Go and replace all the 1's in scientific study with 2's, what do you think would happen? And if you told the scientists that you had done so, wouldn't the action just amount to using a different symbol, and not actually changing the referrent of 1+1=2 at all.

Yes I agree with that. However there is also the question of how scientists are using numbers. They might be using numbers in different ways in different circumstances. They could be using them as merely a labelling system, in which case they would be using the well ordering properties of number. They could be numbers to do account for discrete objects, in which case they using additive properties properties of number. They could be using the numbers to describe proportions or harmonies in which case they would be using the multiplicative properties of number. They could also be using irrational numbers in order to describe spatial distances. The list goes on.

The scientist could talk about things other than number in order to fulfill their needs in any of these scenarios. Its just that mathematicians have designed a multi-purpose tool.

 

[118118]

The scientist could talk about things other than number in order to fulfill their needs in any of these scenarios. Its just that mathematicians have designed a multi-purpose tool

How does the second sentence follow from the first?

 

[Knotted]

How does the second sentence follow from the first?

It doesn't. The negation of the first almost follows from the second. The second sentence is an explanation of why scientists do not use other abstractions than number so often.

 

[118118]

Also my reading of Stanford Encyclopaedia of Philosophy on Husserl indicates that Husserl would not have had any problems with true statements about fictional entities (even including Pegasus!):

“Even objectless (i.e., empty) intentional experiences like your thought of the winged horse Pegasus have content. On Husserl's view, that thought simply lacks a corresponding object; the intentional act is merely “as if of” an object.”

I've thought about this for a while now.

I thought I saw something for a while, but I can't see your reasoning, so I couldn't comment.

 

[Knotted]

I'm not sure what you mean by enumerate enumerations. Its easier to tak in tens. I can count to 500 in tens, or I can count to 500 in ones. I can then count how many times I counted, in tens, or ones. It would seem that we can enumerate enumerations with both ones and tens. I don't see what you do mean by 'enumerate the enumeration'. It seems like a way to crowbar in an alternative mathematics by using simple mathematical terms like 'enumeration' to mean some completely different thing/act.

If you count to 500 in tens then you have counted ten 50 times. In tens this is 5 times. But 5 is not 50. So you have to use a different method of enumeration for the enumeration of the enumeration.

If you count to 500 in units then you have counted a unit 500. In units this is 500 times. 500 is 500. So you do not have to use a different method of enumeration for the enumeration of the enumeration.

I thus eliminated an alternative arithmetic based on counting in 10s. Thus giving a better idea of what we talk about when we talk about 'number'.

I can say that 10 is not a unit because 10xA=A is not true. We need a unit, 'e' say, where exA=A for all A. We denote 'e' by '1'.

 

[Knotted]

So when you say subject matter of mathematics, you mean what mathematicans do, not numbers

No. I don't know what I mean. Mind you not all of mathematics is about numbers. Only number theory is about numbers.

 

[Knotted]

Post 273 should be regarded as a partial elaboration on post 100.

If we have established that we need to define desirable properties of number before we talk about what numbers in general are, then we need to talk about what the desirable properties of number are. I hope this is obvious.

In post 273 I am saying that the notion of unit needs to have the correct multiplicative properties. 10 or 1000 or 253 do not have the correct multiplicative properties so we do not use them. They do, however have the correct additive properties. A geologist can see a rock and say I have here 10 rock and if I add this 10 rock with this 10 rock then I have 20 rocks. Addition works fine in this sort of arithmetic, its just a matter of relabelling - or what mathematicians call systematic abuse of notation (SAN for future reference). But as I say, multiplication does not work after this SAN.

If you are unhappy with the notion of enumeration of the enumeration, then it does not matter as I was only using it to add a bit of colour anyway.

 

[118118]

If you count to 500 in tens then you have counted ten 50 times. In tens this is 5 times. But 5 is not 50. So you have to use a different method of enumeration for the enumeration of the enumeration.

If you count to 500 in units then you have counted a unit 500. In units this is 500 times. 500 is 500. So you do not have to use a different method of enumeration for the enumeration of the enumeration.

I thus eliminated an alternative arithmetic based on counting in 10s. Thus giving a better idea of what we talk about when we talk about 'number'.

I can say that 10 is not a unit because 10xA=A is not true. We need a unit, 'e' say, where exA=A for all A. We denote 'e' by '1'.

I get your last point if eA=A for all e's, then e cannot = 10.

But I don't see why I can't enumerate an enumeration in tens. I have done thousands of times! If I do not forget that I am counting in tens, then I don't mistake the 50 coins for 5 coins.

There is a slight puzzle there as to what is going on in counting in tens, but I'm not going to spend several hours trying to work out why one's brain goes "wait a second" when one looks at that post.

 

[118118]

What about enumeration itself? I count to 50 in tens. 5 is not equal to 50, so...

 

[118118]

I suggest someone else talk to Knotted. I don't want to anymore

If we have established that we need to define desirable properties of number before we talk about what numbers in general are, then we need to talk about what the desirable properties of number are. I hope this is obvious.

E.g. I have no idea how we could have established that.

 

[Knotted]

What about enumeration itself? I count to 50 in tens. 5 is not equal to 50, so...

That's fine. You can count in 10's. I have said this. But why is 10 not the same as 1? I've given my reason. What is your reason?

 

[Knotted]

I suggest someone else talk to Knotted. I don't want to anymore
E.g. I have no idea how we could have established that.

Well if you don't define desirable properties of number then I will keep on inventing new types of number. If you want a loose, undefined notion of number then you will have to accept that there are varieties of notions of number which behave in different ways. Your choice, I'm easy either way.

 

[118118]

I really shouldn't post on this thread anymore, it has swallowed so much of my life. Well, 5 is not the same as 10 because it a different entity/essence/multiplicity/number, whatever.

 

[Knotted]

But I don't see why I can't enumerate an enumeration in tens. I have done thousands of times! If I do not forget that I am counting in tens, then I don't mistake the 50 coins for 5 coins.

You can do an enumeration of an enumeration in tens, but if this second enumeration is also in 10's then your arithmetic will be inconsistent (or rather the law of distribution of multiplication over addition will be violated).

If you count to 50 in 10's then you have counted 5 10's but if you count 5 10's in 10's then you have counted 0.5 tens. 5 and 0.5 are different.

 

[Knotted]

I really shouldn't post on this thread anymore, it has swallowed so much of my life. Well, 5 is not the same as 10 because it a different entity/essence/multiplicity/number, whatever.

I'll leave you be.

 

[118118]

If we have established that we need to define desirable properties of number before we talk about what numbers in general are

But thats not to say we can't talk about numbers before we've defined them. Whatever mathematicans use... I accept all your differnt numbers, just not that they refer to the same things as the numbers I grew up with.

A discussion as to what '1', the mark on a piece of paper, really refers to, the 1 in 1+1=0, or the 1 in 1+1=2, is meaningless; whichever we want it to!

 

[Knotted]

But thats not to say we can't talk about numbers before we've defined them. Whatever mathematicans use... I accept all your differnt numbers, just not that they refer to the same things as the numbers I grew up with.

A discussion as to what '1', the mark on a piece of paper, really refers to, the 1 in 1+1=0, or the 1 in 1+1=2, is meaningless; whichever we want it to!

We need to have at least a very rough idea of what we're talking about before we can classify it.

If you want to talk about whether truth realism about any subject implies ontological realism about that subject then go ahead. I feel you are not so interested in what numbers are than you are in what realism is. But even then we will have to define some sort of properties of realism before we can have any meaningful discussion.

 

[118118]

Just vecause we have to share a rough_idea of what numbers are does not mean that we have to *define* numbers. I have yet to come across a definition of numbers, but I have read a *few* pages on the philosophy of them.

I am intersted in what a sustainable realist theory of numbers would look like.

I'm not sure why we have to scetch an outline of realism before we "discover" what realism wrt numbers would look like... but mind independent seems like the only catch all description.

 

[Jonti]

I'm not so sure about that, to tell the truth. How could we tell whether mathematics is somehow embedded in the world, implicit in its workings; or whether it is embedded in any perception of that world?

If math is an inevitable feature of mind, that would be a form of realism.

 

[118118]

Well not according top this book. I'm not going to post on here again, anyway. No-way!

 

[Jonti]

Originally Posted by Jonti

I think the suggestion is that the (perception of the) passage of time implies two-ness because this is now, and that was then. But, to my mind, it's the perception of difference that's really fundamental. It is only by making a distinction that mathematics become possible; and that implies a cleaving of a previously unified space.

The first distinction, one might say, creates duality.

As I understand it Brouwer thought that time is synthetic a priori but space is not. I'm not sure why he thought this, but it seems to be why he emphasised temporal difference rather than spacial difference.

Incidently this line of thinking seems to be related to some of the things in the Revising Reality thread. Our brodix and Lee Smolin seem to be getting all Kantian about time. Maybe Brouwer is another philosophical kindred spirit.

Yeah, I've a lot of sympathy for that, but I think the case is made *much* clearer by leaving out quite a few of his words ...

"Mathematics arises when the subject of two-ness ... is abstracted from all special occurrences. The common content of all these two-nesses becomes the original intuition of mathematics and repeated unlimitedly creates new mathematical subjects."

The common content of all two-nesses is exactly "a distinction". The argument seems to be that maths is the science of distinctions, the science of signs themselves. As we, or any sentience, cannot help but see the world with distinctions, and to think about it with signs, mathematics appears as a universal real, ubiquitous in the world.

 

[Knotted]

The comment content of all two-nesses is exactly "a distinction".

I think it is more specific than that. "Well ordered distinction" might be closer. Especially considering time is brought into into it. Things in the past might not be so much distinct from things in the present but they are certainly 'before' and things in the future are certainly 'after'. This will give an intuition of less than and greater than. The ordering intuition is perhaps not so intuitive when it comes to spacial distinctions - you have left and right but one might as well be the other.

 

[Jonti]

Oh, yes. I think it's well worth quoting in full ...

Here's a point of view from Luitzen Brauer where 2 seems to be fundamental:

"Mathematics arises when the subject of two-ness, which results from the passage of time, is abstracted from all special occurrences. The remaining empty form [the relation of n to n+1] of the common content of all these two-nesses becomes the original intuition of mathematics and repeated unlimitedly creates new mathematical subjects." (quoted in Kline 1972, pp. 1199-2000)

I'm happy to accept that we find time as a given, and that as a given it is prior to space (or, to be pedantic, any particular perceptual space). Within a period or duration of time things are ordered, sequenced.

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. Every sensory space, of any possible sort, carries with it the notion of intensity. Its content may dim and grow faint. Light may be more or less intense; as may sound, smell, or whatever.

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

Hmm. Nice. And I can't help but notice that number is right there at the heart of Brouwer's original intuition of mathematics

 

[Knotted]

Oh, yes. I think it's well worth quoting in full ... I'm happy to accept that we find time as a given, and that as a given it is prior to space (or, to be pedantic, any particular perceptual space). Within a period or duration of time things are ordered, sequenced.

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. Every sensory space, of any possible sort, carries with it the notion of intensity. Its content may dim and grow faint. Light may be more or less intense; as may sound, smell, or whatever.

It seems to me that ordinality, or ordered sequence, is given to us by time; provided one can distinguish successive moments. 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'm not sure that Brouwer would have agreed with that. Don't confuse the notion of magnitude with the notion of number, especially when discussing intuitionism. A magnitude is a point in the continuum and thus is not usually finitely expressable. Intuitionist mathematics, on the other hand, is entirely expressable in finite terms (which is not to say that infinities are not considered, just that the construction of the infinities must be expressed in terms of the finite).

Here's a similar, but better (IMO) quote from Brouwer:

"Completely separating mathematics from mathematical language and hence from the phenomena of language described by theoretical logic, recognising that intuitionistic mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time. This perception of a move of time may be described as the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory. If the twoity thus born is divested of all quality, it passes into the empty form of the common substratum of all twoities. And it is this common substratum, this empty form, which is the basic intuition of mathematics."
http://www.marxists.org/reference/subject/philosophy/works/ne/brouwer.htm

The above had a big influence on my thinking on this topic - even if I reject the Kantian flavour and I don't subscribe to intuitionism. I think that Wittgenstein's notions of family resemblance are a better starting point than Kant's categories of pure thought.

Hmm. Nice. And I can't help but notice that number is right there at the heart of Brouwer's original intuition of mathematics

I certainly like that bit that and the idea that the 'central intuition' is languageless/logicless.