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A letter from a student *June 17, 2008*

*Posted by Alexandre Borovik in Uncategorized.*

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Hello Professor,

I hope everything is good and that you recovered from the accident in your

finger.

I just wanted to share some personal thoughts. Suppose we have a system that is discrete and finite. We have the natural numbers {1,2,3,…,T} where T is the symbol for the ”biggest natural number”. We will never have a value for T but we accept that it is a fixed natural number. We can also include 0 in our system. How much mathematics can we do?

We could define the usual addition and multiplication. But we will have a problem when the result is ”greater than T”. But nothing ”greater than T” exist…

Suppose for example that T=10. Then we have 2+3=5, 4+5=9, 5+5=10. But what about 5+7? We could just define 5+7= err (error). just like the small calculators do when they reach their limit. But err is not in our system… We could work modulo T OR we could just say that 5+7=10=T. and 8+9=T. and 7+8=10=T etc.

(If we choose the last option then obviously 1+T=T and T+T=T. So T has a similar behaviour as the infinite that we learned at school. But the big difference is that in our system T is a fixed natural number!)

I just can not see why we NEED infinite to make mathematics. Is it a matter of convenience? Is it just to make things simpler? If this is the case then we should accept that infinite and continuous entities are just tools, ideas, symbols that make our life easier. But we should remember that IDEAS themselves are FINITE since they live in the finite world of our brains!

I think the whole problem is more about aesthetics. I think that someone can accept that only discrete and finite things exist and, at the same time, that this belief does not destroy all the beautifull mathematics that have been created until now.

While I was searching the Internet I found this PDF document which I attached and I found amusing. I also found other people expressing similar views (against infinite) like for example Alexander Yessenin-Volpin.

[…] A letter from a student If this is the case then we should accept that infinite and continuous entities are just tools, ideas, symbols that make our life easier. But we should remember that IDEAS themselves are FINITE since they live in the finite world of our … […]

I’m with you, student (just finished a degree in Maths and Computing).

Mind you, if some people want to play in that particular field, I have no objection per se.

I do think that it should not be considered as mainstream for (almost all:-) ) educational purposes.

One problem is that there are combinatorial statements (I’m told by combinatorists, I can’t directly give you an example) where in order to prove a statement about strictly finite objects, you MUST go through an infinite set. So then there ARE statements about finite collections which can be proved now but which couldn’t be without the infinite. So we CAN’T do even all of the finite math that we currently can if we were to abandon infinite sets. It’s not just a convenience, infinity is a necessity.

2 Charles

You can always skip the step of interpreting all these “infinite” things as infinite, and just carry out the formal rules. This, in particular, means that you can construct another proof, that is equivalent to the one you know containing infinities, that has “opaque-but-useful-symbols” instead of “infinities”. In the worst case, you’d only need to recast some axioms in the new form. Infinity is a useful term to use as an intuition-pump, up to some point, just as you use word “point” and not “tniop” to describe elements of metric spaces.

In many cases (eg, the use of probability theory in statistical mechanics; mathematical finance; epidemiology), the infinite is used as an approximation to the very-large-finite. (Yes, you read that correctly!) This is because the mathematics of problems involving infinite numbers of entities is often more tractable than for problems involving very-large-finite numbers.

Indeed, the differential calculus is another case where the infinite is easier to handle mathematically than is the finite. The common rules for differentiation or integration of functions would be a lot harder to derive and to remember if they were applied only to functions defined on finite subsets of the number line, for example.

Maya, I think one _should_ also talk about this in educational settings.

There are people who do not see through the formalism, and they think that they are really handling infinites in their brain; whereas of course they are only handling quite finite symbols (with iteration as an intuition pump for “reaching” infinity).

This is important when applying math to physical settings: physicists often use mathematical infinity in theories as an argument against digital physics approaches.

http://en.wikipedia.org/wiki/Digital_physics

So, not talking about it in education could stop grad students from entering certain research areas.

This is just a lexical mistake.

Everyone is using the term “natural numbers” to refer to the set of objects to which you ‘add one’ to obtain the next object, starting from zero. If you decide to name it another way it still remains the same concept. And all theorems remain true for it, whatever name you put on them also.

Like the number one, you can decide that actually it is two. But what’s real in that ? Where did you mean to give a name, and where did you mean to refer to what this name represents ?

These concepts aren’t just names we put next to each other, there is an essence behind mathematical objects. If you are to consider a system of natural numbers {1,2,3,…,T} then just write it as [1..T] and call it a range of integers. As this kind of mathematics is a pure theorical science, the true essence of its objects precedes everything else, including the names we put on it.

My conception of “natural numbers” is that they start from zero, and you can build the next number by adding one. But I am talking about the concept, not the name, not the description phrase, not the representation of it with marbles, or digits on a blackboard. Pure mathematics are abstract.

If we share that definition for “natural numbers” it is clear that there is no such thing as a “biggest natural number” as you were considering since for each T we can construct T+1.

So at this point you were asking “how much mathematics can we do ?” and I would harshly say “none since you did not even get what is a natural number.”

However it was an entertaining pseudo-subversive post, just like the Ramones and music, except that they probably drank much more than you in the process =:P

Maybe you still care about the rest of the discussion : we dont NEED infinity. You are only a massively crippled thinker if you refuse to consider such a thing. You would also be free to ride a bridge without considering it is not finished. We don’t NEED to live actually.

Mathematics is not about believing, it is about defining some rules (the axioms) and give a name to what we can observe given those rules (conjectures, theorems, definitions). In our case the term “natural numbers” lay the rules and anyone not so blind would observe something he would call “infinity”.

“Everyone is using the term “natural numbers” to refer to the set of objects to which you ‘add one’ to obtain the next object, starting from zero”

Well, that’s the problem really, this set (an infinite set axiomatically defined to exist) can’t have 1 added to it, can it?

Computable enumerability, constructivism, digital physics etc suits me just fine thank you very much (and most other people I would venture).

Anyway, I can do everything I need to do without engaging this concept and I don’t feel crippled in the slightest:-)

I still maintain that this kind of thing constitutes research for a relative few and ought not to be baked in (presented by default as the only truth) to mainstream education.

Thomas, your own argument cuts your legs out from under you.

What the student is asking about here is the possibility of defining axioms that (a) entail a finite atomic ontology, and (b) are sufficiently powerful to do

foo. Is there a finitistic system which is powerful enough to count to ten? Sure: just take the natural numbers less than or equal to ten. What about to write out the standard elementary school times tables? You just need 144 for that.What if we say our system’s ontology only contains atomic referents. Then what can we do with it? Clearly we can count to . We could multiply any numbers as long as their product didn’t exceed that value. Or maybe we could make alternative interpretations as signed numbers. Or maybe “floating-point” representations of a usefully large collection of fractions. What system of simple axioms would let us do the most with our limit of ontological slots?

I’d say it’s a massively crippled thinker who needs infinity to count to ten.

If I may, I think the more reasonable position is one of agnosticism rather than theism or atheism. That is, rather than assume the Successor Axiom SA (that every natural number has a successor which is a natural number) or its contrary (that there is a maximum natural number which has no successor which is a natural number), one should simply work in a system which makes no assumption either way.

So consider FPA, second-order arithmetic minus the successor axiom. This system has models the standard one (if one believes it exists) and all initial segments (however many there are). In this system, if one is given a natural number, then one cannot prove that there are any larger natural numbers, but one can prove that all smaller numbers exist.

So what can one prove? Well quite a lot. Interestingly, the system can prove its own consistency (which is not contra Godel, because the systems the 2nd theorem refer to assume SA).

It can also prove Quadratic Reciprocity and Bertrand’s Postulate and, I imagine, Fermat’s Last Theorem and a version of the Prime Number Theorem.

http://www.andrewboucher.com/papers/arith-succ.pdf

http://www.andrewboucher.com/papers/bp.pdf

Regards

Thomas said; “

My conception of “natural numbers” is that they start from zero, and you can build the next number by adding one.”Our intuitions differ a great deal, it seems. Just what is “natural” about the so-called number zero? How can zero be a counting number, since the activity of counting (by definition) presupposes there are objects to count. Zero objects cannot, by definition, be counted. So, any collection called “the natural numbers” can only start from one, not zero.

You’ll probably next try to persuade me that combining a minus sign with a natural number, as in “-3”, is also some sort of number. Fooey! I’d like to see you try to count one of those!

Just what is “natural” about the so-called number zero? How can zero be a counting number, since the activity of counting (by definition) presupposes there are objects to count.Cardinality of empty set? (Or do you think there is no such thing as the empty set?)

If you don’t think zero is natural, try defining addition from the Peano axioms without it.

In this (and

manyother algorithms), iterations count down to zero. There’s a reason assemblers include the fundamental “branch if zero”, and not “branch if one”.14: Perhaps I am misunderstanding you, but AFAIK there is no difficulty in defining addition from Peano Axioms without zero.

x + 1 = x’ and x + (y + 1) = (x + y)’,

where ‘ is used to indicate successoring.

0 is unnatural to this extent. Consider second-order Peano Arithmetic without the Successor Axiom. So (again) this has models the initial segments, as well as the standard model. Then one can prove the following version of the existence of infinity:

(there exists P)(for all natural numbers n)(P is not n in number).

Note that this still works in initial segments. E.g. in the initial segment {0,1,2,3,4}, the universal set (i.e. {0,1,2,3,4}) is 5 in number, so cannot be numbered by any finite number in the universe. Of course, it can be numbered by a finite number (5) outside the universe. This is akin to Skolem’s Paradox.

When starting from 1, one cannot prove the same result.

Zero is not natural, even less so is the empty set.

A proof that any two empty sets are equal usually involves the use of the outrageously infinitistic Axiom of Extensionality. To many students in my Foundation Year course (and they represent countries from Afghanistan to Zambia) a proof of the uniquiness of THE empty set is a revelation — not so much mathematical as linguistic, because for many of them their native tongue has no definite article, or usage of definite articles is different from that in English. In some languages, definite articles can be used only to point a finger to a concrete object and could be best translated not as “the” but “that”. Try to say “take that empty set”.

Peano axioms without zero are used, for example, in the classical textbook by Edmund Landau “Foundations of Analysis”.

abo (yes, I see that you are a name and not just a comment number): Great for pattern-matching, horrible for algorithms. How, pray tell, are you to recover given ?

To both you and Alexandre: I didn’t say that it’s impossible to give an algorithmic definition without zero. I just assert that the result looks very sloppy. Multiplication gets worse.

Zero is not natural, even less so is the empty set.I am really surprised by the dogmatism of this statement, and I really disagree. Is then the notion of falsity also ‘unnatural’? The idea of nothing?

Perhaps I don’t understand what you mean by ‘natural’.

The literal uniqueness of ‘the’ empty set (the set of pink unicorns = the set of humans living on Mars) is somewhat irrelevant in my opinion. What really counts (ha ha) is the presence of an initial set: a great convenience for many, many reasons. And initial objects are unique up to unique isomorphism, which for any mathematical purpose is just as good as (actually, more natural than) literal uniqueness. Similarly, one says ‘the’ disjoint union of two sets, even though such a thing is not literally unique, but only so in this categorical sense.

I’ll say this: the empty set is a great convenience, so that it would (to me) be awkward and unnatural to do mathematics without it. It’s ‘second nature’, then. I think John Armstrong is saying much the same thing.

Todd: that’s more or less it. A lot of things look really ugly without it, so it seems natural to use zero.

Overnight I thought of another note in response to Alexandre’s comments about the empty set. I think Todd would be behind me in saying that the definite article is actually the least important thing here.

If you’ve got extensionality, then yes you have actual uniqueness. But Todd and I would both be perfectly happy to throw it away and make do with uniqueness up to isomorphism (which does

notrequire extensionality).John: “How, pray tell, are you to recover y given y’?” I don’t understand this question, nor how it imports to the issue of the naturalness of 0.

” I just assert that the result looks very sloppy. Multiplication gets worse.” You can, of course, just assert it. But still…

In Peano Arithmetic with 0, the definition of addition is

x + 0 = x

x + y’ = (x + y)’

and the definition of multiplication is

x * 0 = 0

x * y’ = x * y + x

In Peano without 0 the definition of addition is

x + 1 = x’

x + y’ = (x + y)’

and the definition of multiplication is

x * 1 = x

x * y’ = x * y + x

So why is the definition of addition sloppy? And by what measure does multiplication get worse??

Todd —

I merely said that I thought zero was not a “natural number”. I said nothing about doing mathematics without it, which is a different and completely unrelated issue. We use all manner of artifical constructs in mathematics, as in the rest of science and engineering, and so we should. We can even use the concept of zero without zero thereby becoming a natural number.

Peter —

It goes without saying that you are welcome to your opinion. But just to review: you said zero was not a counting number, to which I replied in comment 13 by suggesting that zero is the counting number for the empty set. My comment 18 was in reply to something Alexandre wrote.

But, since you’ve brought it up again, I’ll say a bit more about this. In dealing with sets, it is a completely natural thing to construct sets by applying the principle of (bounded) comprehension: given a set X and a predicate P defined on X, we can form the subset {x: P(x)}. In some cases, this may yield the empty set.

For example, if X is the set of positive natural numbers and P(x) is the predicate “x is an odd perfect number”, then it may very well be that this subset is empty; we don’t know yet. It would be perfectly natural for a mathematician, while conducting research on this question, to contemplate the nature of this set. That’s completely standard practice. If in the fullness of time she manages to prove that the set is empty, does it suddenly become something “artificial”?

In a similar vein, I asked Alexandre whether he thought the notion of falsity were also unnatural (or artificial, to use your word). Well?

If you accept that the empty set is not such an unnatural construct after all, isn’t zero just as natural?

(I’m not absolutely sure of this, but I believe the convention of most mathematicians these days is to include 0 when defining the set of natural numbers.)

Perhaps the student would enjoy Jan Mycielski’s 1981 paper, Analysis without Actual Infinity from the Journal of Symbolic Logic, vol. 46 p. 625. Mycielski uses a system with axiom schemes whose models are all infinite, but with the property that each finite collection of axioms has a finite model in a straightforward way, by replacing certain constants with finite natural numbers that are large enough to do the job.

[…] often bandied about by mathematicians, is perhaps an overloaded term (see the comments here for a recent disagreement about certain senses of the word). I don’t know the exact history […]

Perhaps to reconcile views on ‘naturalness’ we can give a similar interpretation to that given by Polanyi for simplicity in the long quotation at the end of this post. To be mathematically natural is not to be obvious or easily seen by a lay person – recognition is only gained by years of training. But it does point to what’s rationally appraised in the discipline.

Todd —

you move from a discussion of naturalness to a discussion of mathematical naturalness. I see no reason to suppose that these two terms are the same. Indeed, doing mathematics itself might well be considered by some to be perverse and unnatural behaviour, even though most enjoyable!

Indeed it might, although I think it’s you who’s now shifting the discussion — I thought we

weretalking about mathematical entities (on a basic human level, even).