> What we call "laws of nature" are interpretations of what we think
> these laws are. These interpretations vary with culture, with the
> development of mathematics, with what we are willing to accept that a
> "number" can be, with the domain to which they are applied, etc.
> Because they are necessarily partial, it is not really accurate to
> call them "laws" though doing so seems to comfort a lot of people.
Of course the meaning of "law" in "scientific law" is a metaphor, which originated, I suppose, in the old idea that they were ideas in God's mind when he created the world, and which he "imposed" somehow on his creation. But both empirical scientists and mathematicians have no reluctance to use the world, so I don't see why it creates any problem, as long as we can keep in mind what we are actually talking about.
> Well, in base 3, the argument wouldn't be false, it would be
> nonsensical.
I hope you realize, as John Kozak pointed out, that modulo arithmetic is *part* of ordinary arithmetic, not an alternative to it. So your example is not in any sense an example of an alternative mathematics.
> Moreover, the truth of science isn't given by the verifiability of
> 2+2=4, which is true by definition. One possible "truth" of science
> has to do with what matters numbers can be applied to and to what
> effect. The idea that "mathematics" is the language of nature was born
> in the sixteenth century and has now become part of "common sense."
Well, it goes back to Pythagoras, at least, so it's a lot older than the 16th century. But you are getting close to an important part of this whole subject. That is, why does mathematics proves so useful in dealing with the physical world?
There is still no consensus in the field of philosophy of mathematics about this, and I'm far from a specialist in that field. But if I had to make a guess, it would be along these lines.
Mathematicians love to formalize things like arithmetic and geometry. They define certain terms, make up axioms using them, and derive theorems from the axioms using logical rules. Once you have a formal system, you can look for models for it. It so happens that the formal systems of arithmetic and geometry can be modeled by many objects in the physical world. As Carroll is insisting, there aren't any "perfect" circles, etc., in that world, since there aren't 0-dimensional points, 1-dimensional lines, etc. (Maybe there are some lurking somewhere -- who knows? -- but we'd have a lot of trouble finding them?)
It turns out that individual objects (fingers on my hand, coins in my pocket, etc.) are very good models for the integer part of arithmetic, and manhole covers, wagon wheels, etc., are very good models for the circles of geometry. So we can apply the theorems of arithmetic and geometry to these objects, which is not surprising, since those objects were themselves the inspirations for the formal mathematical systems.
There is a lot more to this story, but I have to get back to work now, so if you want me to go on, I'll have to do it on a later date.
>> there's also Feyerabend of course,
>>
>> Most philosophers of science don't think much of him, and neither do
>> I.
>
> Well, there we differ.
It's not just a question of differing -- people differ all the time in philosophy. It's a question of who has the stronger argument, when arguments are pitted against each other in free discussion. Most philosophers of science who have had extensive experience in the subject think his arguments are weak, I would say, and I agree with them.
>> Cristopher Caudwell (The Crisis in Physics, Illusion and Reality),
>>
>> I know the name, but haven't read anything by him. I don't think he
>> has much stature in the philosophical world, though.
>
> Ah, we're talking about stature now? I thought we were talking about
> truth.
We are. The reason I said that was that, as far as I know, not many philosophers (professional, that is) have discussed him. In any case, I don't have time to look him up and study his theory of mathematics right now, so if you would like my opinion of his ideas, would you provide me with some sketch, at least, of what they are?
> Think about the following:
>
> --Try to multiply 1,435,763 by 329,548 in Roman numerals.
It can certainly be done, though it is much harder than using Arabic numerals.
> Do you think the notation system affects our notion of mathematical
> possibilities? If the answer is yes, what do we make of the
> essentialism you attribute to mathematics?
It certainly doesn't affect the result of multiplying 1,435,763 x 329, 548.
> -- What kind of a mathematics do you think we could develop if we only
> admit integers as genuine numbers (the Platonic position).
It would be integer arithmetic -- a subset of the mathematics we have, not an "alternative" to it.
> -- What kind of mathematics do you think we could develop without 0?
Well, some people got along without it for quite a while, but once again, they were only able to use a subset of the mathematics we have, not an "alternative" to it.
> -- What kind of mathematics will result if the reality of a number is
> given simply by our ability to represent something as a possible
> number. For example, what kind of number is x in the equation x = the
> square root of (-1)? Is it a number like 1, 2, 3 .... is a number?
It's i, a perfectly good number in the mathematics we have. Look, Joanna, I appreciate your efforts, but you haven't come up with an alternative to mathematics, because there is *only* mathematics. I have to run now, so catch you later.
Jon Johanning // jjohanning at igc.org __________________________ In all ... philosophical studies, the difficulties and disagreements, of which its history is full, are mainly due to a very simple cause: namely to the attempt to answer questions, without first discovering precisely what question it is which you desire to answer. -- G. E. Moore