For it is necessary, if we are to be justified in accepting this or that as an axiom, that its logical consequences not conflict with what is already known about the numbers, but instead include things already known about the numbers.
So we cannot be justified in accepting the axioms unless we already know a lot about the numbers independently of the axioms or any deductions from them.
After all, its not as if mathematical knowledge began with Peano!
C. Your idea seems to be this: Lots of things get to be known about the numbers. Then someone proposes certain sentences as axioms. So the question arises, ought we to accept these sentences? And we decide whether we should by seeing (i) whether they logically entail antecedently known arithmetical truths, and (ii) whether they logically entail anything inconsistent with antecedently known arithmetical truths. Only if they entail what we already know, and entail nothing which conflicts with what we already know, are they acceptable axioms.
Isn't this your thought?
M. But the statements we accept as axioms were among the ones already accepted as true.
Peano's proposal was not that certain sentences (i)are true and (ii) may serve as axioms. Rather, he selected certain already known arithmetical truths and proposed only that they may serve as axioms.
Peano was not the first person to suppose that zero is the first of the natural numbers or that different natural numbers have different successors! His innovation lay not in asserting the things his axioms assert - as mathematical assertions his axioms were already familiar and accepted - but in proposing that these statements were logically sufficient for the whole of the arithmetic of the natural numbers.
P. Fair enough. But my point still holds good: we can be confident of the truth of the axioms only if their consequences, as so far known, agree with what we already know about the numbers, both in the sense of conflicting with nothing already known and actually leading to things already known.
For suppose someone were to propose as an axiom some sentence which logically entails something which goes against some known mathematical truth. Wouldn't we then reject the proposed axiom?
C. I find the question virtually meaningless.
Suppose that some particular equality - for example that 5+7=12 - logically entailed something which went against some piece of mathematical knowledge already in our possession. How would you answer the question about what we would then reject?
Wouldn't you rather become suspicious of the question, or the supposition upon which it rested?
P. I don't follow you.
C. Well, we are supposed to suppose that we discover that the statement that 5+7=12 logically entails something which conflicts with something we already know. So suppose, if you can, that it logically entails that 4=5.
Now, what gives way? Do we reject the equality 5+7=12 or the inequality 4≠5?
Suppose we reject the inequality. Then we agree that 4=5. But 4+0=4 and 4+1=5. Thus, since 4=5, 1=0. But then, since 2 comes right after 1, 2 comes right after 0. But 1 comes right after 0. So, 1=2. So, 0=1=2. And thus also 0=1=2=3=4=5=6... That is, then all the numbers are one and the same!
Well - that makes it look as if we'd better reject the equality. Then we must hold that 5+7≠12. What then is 5+7? Well, it must be greater than or equal to 7. Suppose it is 7. Then, 5+7=7, in which case 5+7=0+7. But then 5=0. But since 6 comes right after 5, 6 comes right after 1. So 6=2. By the same reasoning 7=3 and 8=4 and 10=5 and 11=1, and so on. That is, all the numbers are just the numbers 0 through 4. Now suppose that 5+7=8. Then 5+7=1+7, in which case 5=1. But then 4=0, in which case there are just the numbers 0 through 3. And so on.
So the situation is disastrous on both alternatives.
P. I see that.
M. The same sort of thing arises in respect to the «supposition» that some axiom entails the equality that 4=5, for example the axiom that zero is the first number. What would we reject if it turned out that if zero is the first number then 4=5? If we reject the axiom, then we agree that some number comes before zero. But then zero comes immediately after some number. So let the number be one. Then zero comes right after one. But so does two. But no two numbers come right after any number. So, we conclude that 0=2. And if that is so, then 0=3=4=5=6=7=8=9 ... just as well. And if it isn't one that zero comes after, it is some number - and the conclusion will be that the numbers then stop with its immediate successor, namely zero!
So, if we reject the axiom we will have to accept something outlandish, such as 0=2. If we do not reject the axiom we will have to accept something equally outlandish, for example that 4=5.
So, and please do not take this remark badly, it seems to me that no thinking has gone into this idea that we must test the axioms against what is already known. It has just been words without thought.
P. Perhaps there is something in what you say. I do now feel as if I never bothered to think through the supposed supposition I was making. But what if someone proposed something brand new as an axiom - then, surely, it would have to meet the test of agreeing with what was already known in arithmetic, and so couldn't be a source of knowledge.
C. But what is this «something brand new»? Presumably it will be a sentence of arithmetic. So it won't be syntactically or semantically novel. So it must be one whose truth-value is as yet unknown to us.
But no one proposes as an axiom for arithmetic some arithmetical sentence we aren't already sure of!
P. But surely this could be done. Didn't Gödel suggest that we might try to think up further axioms for set theory? And any such axiom would not really be known to be true at the start?
C. Perhaps. But right now we are talking about axiomatic arithmetic, not axiomatic set theory. The two may be quite different, and certainly the intellectual pressures that gave rise to them are entirely distinct. So let's just stick to arithmetic. And here my point is this: In arithmetic no one proposes as axioms statements about which we are as yet uncertain.
P. As I reflect on what you say, I find myself forced to agree. The actually proposed axioms, and very likely any that anyone would seriously propose, are statements about whose truth we are already satisfied. In a word, and this is the point I did not appreciate, axioms are selected from what is already known.
I think that what happened in my thinking was that I slipped from asking what would be needed to accept a statement as an axiom to asking what would be needed to accept the statement. Then, since a condition on its being an axiom is that a statement, together with the other axioms, leads to all the already recognized truths, I slipped into thinking that this was a condition on accepting the statement itself - as if it needed to pass a test taken simply as a statement of arithmetic!
In any case, I now seem to see that it is quite correct to say that the statements which are proposed as axioms for arithmetic need pass no special test since they are all among the statements already known to say true things about the numbers.
But though I will grant that proposed axioms state about the numbers things already known, surely there is much in addition to what is said by those axioms that also is already known about the numbers. And those further, already known things are not known by deduction from the axioms. That point remains secure.
And so I will continue to assert that the axioms and deductions from them are epistemically irrelevant.
C. But if other things in arithmetic could be known by deduction from the axioms, then if we can see how we do know that the numbers are as the axioms state them to be, then we can see how we can know all the rest. We could gain a certain insight into the knowability of the propositions of arithmetic, even if not into the manner in which all or some of them have actually become known.
P. But it well may be that we couldn't know that the numbers are as the axioms state them to be without already knowing ever so many of the more specific points about particular numbers which follow from the axioms. And in that case we will not gain the insight to which you refer.
I will make my point this way: You assume that the axioms can be known on their own and apart from knowing anything else pertaining to the numbers. But this assumption may be in error.
C. Is that my assumption? It may be a kind of psychological impossibility to know that zero comes before all the other numbers and not know that it comes before one. For it might be impossible for us not to see that zero comes before one if it comes before all numbers other than itself. If so, we could not recognize that zero comes first without recognizing as well that zero comes before one. So it might be impossible to know the truth of the general proposition without knowing the truth of some of its instances. I have not suggested or assumed the opposite.
But that would not show that knowledge of the truth of the general proposition is based on knowledge of the truth of its instances.
P. On what else might it be based? Can't we see that it must be based on knowledge of the truth of at least some of its instances?
C. I am not at all sure on what our knowledge of the truth of the axioms is based. But I think I can see that it is not based on such knowledge of the truth of its instances as we may possess. And here is how I see the matter. Consider the axiom which states that zero precedes every other number. Its instances are `zero precedes one', `zero precedes two' and all the rest. Now, we know the truth of only finitely many of these instances. But we surely couldn't know that zero precedes every other number by knowing that it precedes three or fifty or fifty thousand other numbers. So our knowledge of the truth of the axiom cannot be based on our knowledge of the truth of its instances.
If we came to believe that zero precedes every other number by noting that it precedes one, and two, and three... and five million, then that would be merely a conjecture - not a piece of knowledge. And a very weak conjecture at that, since we would be extrapolating from the finitely few to the infinitely many.
Yet, we surely do know that zero precedes every number other than itself. And so it seems that this knowledge is independent of such knowledge of the truth of its instances as we may possess. And the same holds for each of the other axioms. So they all are known independently of their instances. But their instances can be deduced, and thus known through them.
So that is one way in which knowledge is possible in arithmetic, and this way of knowing will become fully clear to us as soon as we make it clear to ourselves how we do know the truth of the axioms.
P. I feel very uncomfortable about this. It continues to seem clear to me that we know e.g., that zero does not come after one, and know that every bit as much as we know that zero follows no number and do not know it by deducing it from that general proposition.
C. But I have said only that by reference to the axioms we can arrive at an understanding of how arithmetical knowledge can be obtained - not that we will arrive at an understanding of how arithmetical knowledge must be obtained, or even has been obtained.
It may be that we can know that zero comes before thirteen by deducing that proposition from the known truth that zero comes before every other number, and that is consistent with our actually knowing that zero comes before thirteen independently of any such deduction.
I am not suggesting that we ordinarily do know, much less than we must know, the familiar sums and products by deduction from independently known axioms. I say only that we can know these sums and products by deduction from independently known axioms.
P. I see what you're saying, and I don't see how to object to it. Yet there seems to me to be something wrong in your conception of the matter.
It seems so clear to me that knowledge of the truth of various instances comes first and that knowledge of the truth of the generalization comes second and couldn't come first. And so I feel that knowledge of the truth of the axiom really is secondary or derived. Despite your arguments, I cannot shake myself of this feeling.
C. Well - you say this, but then you don't explain it. So I am quite at sea about how best to respond to you. I really don't see why you say we couldn't know the truth of some consequence of an axiom by deducing it. Deduction surely is one way of obtaining further knowledge from some already obtained knowledge. I hope you will not go so far as to question that.
P. That has tempted me. But let me try this instead.
As I understand it, people had a good grip on a lot of arithmetic prior to getting a handle on zero. This shows that people could know - did know - enormously much about the numbers without so much as conceiving the content of most of the axioms - for most of them include the idea of zero.
C. Agreed. But that would be a somewhat different system of arithmetic, and so would have somewhat different axioms. Instead of asserting that zero comes first among the numbers, the right axiom for this arithmetic would assert that one comes first among the numbers.
But what turns on your point? I have not denied that our actual knowledge may be independent of deduction. I have only asserted that deduction from antecedently known axioms is a way of knowing.
P. But the important point is that it is possible to have fragmentary arithmetical knowledge. And just as there is arithmetic without zero, there can be arithmetic without generalization. People can know a lot about individual numbers - their sums and products - without as yet so much as considering generalizations like `Zero comes after no number'. It might not be the case that anyone who considers that statement fails to recognize its truth. But people might be, as it were, oblivious to it and still know lots of things about lots of numbers.
So I will now put my point this way: Our knowledge of the individual numbers - as contrasted with our knowledge of arithmetical laws - in no way depends upon our knowledge of those laws.
C. But now you are no longer viewing the epistemic irrelevance of axiomatic arithmetic in the same way that you did at first.
P. I agree. My initial «view» made no sense at all. It was, as you put it, words without thought.
But that recognition did not free me of a feeling that somehow the laws of arithmetic are secondary. And so, I am now working on that idea.
What I now wish to say is that the fact that we can know enormously much about the numbers without so much as considering the laws shows the irrelevance of those laws to our basic arithmetical knowledge.
C. In a way what you say seems undeniably correct. But aren't you still missing my main point - for I have only said that it is possible to arrive at arithmetical knowledge by deduction from axioms independently known to be true. Not that we must or even actually do so arrive at arithmetical knowledge.
P. Yes. For some reason I keep slipping up on that point. Still - there is something I am after, so let me shift ground again.
What I now want to say is that knowledge of an axiom does depend on knowledge of its instances. Not all of them, but some of them. We first come to know instances. We perceive a pattern in the instances. We then generalize on the known instances to bring out that pattern. I do not mind calling that a kind of speculation. Then by deduction we bring out further particular points that we can then put to the test.
This, I now think, is how things work in arithmetic. We know the axioms to be true only by seeing that again and again their entailments are true.
C. Here you're viewing the axioms as analogous to expressions of empirical regularity. This much is right in that analogy: the axioms are laws. They are laws of numbers.
You're also suggesting that the laws of number are hypotheses suggested by particular facts and then tested by seeing what further particular facts they lead to.
This makes knowledge of the laws entirely secondary.
But I feel that in some way knowledge of laws is present in even the most elementary pieces of arithmetical knowledge - in a way in which knowledge of an empirical generality is not involved in knowledge of its instances.
I could put how I feel about the matter this way: I need not in any way work with the idea that all ravens are black to spot the blackness of this raven.
So I just cannot accept the sharp distinction you draw between knowledge of the laws of number and knowledge of the numbers.
P. That is extremely obscure.
C. I agree. And I could say even more obscure things than that on the topic! But don't worry about the obscurity, for even an obscure remark might lead to better things.
So let me go on and try this assertion, that there is something general involved in what goes into knowing anything arithmetical.
P. I would say that you were true to your word, you can indeed increase the obscurity of your pronouncements!
But seriously, tell me what you have in mind - if you do have anything in mind.
C. Well, something general comes into learning the numbers - learning which numbers are which and what the sums and products are. For you don't just learn this number and then that one, in a random fashion, as if you were learning about this raven and then about that one, but you learn a system. You learn to go from one to two, and from two to three, and so on. You need to do something like master a rule. And there is something general about a rule.
I would not deny that this is still very obscure. But it feels right nonetheless, and perhaps just a bit less obscure than what I began with.
P. But don't some people count one, two, many? Or a child might master just the numbers through five.
C. Yes. There are these fragments of arithmetic which, taken on their own, are in some ways like and in more ways unlike arithmetic. But if you've got the numbers then you don't have just five or fifty of them. In a certain sense you have them all. And that's where the generality or law comes in. We might say that you master a law of construction for the numbers, or for recognizing them. And it is the mastery of that law which gets reflected in the first two axioms.
A person might be thrown by the question `And is there a greatest number?' while yet knowing full well that if there is one more swan than fifty there are fifty one swans, and if there is one more swan that fifty thousand, then there are fifty thousand and one swans. And so forth.
And the person thrown by the statement that one comes before every number other than itself will nonetheless always start with one when counting! Here, we might say, a certain feature of his or her practice constitutes a kind of recognition that one comes before all the other numbers - and it is this feature of practice that is explicitly set out in the axiom.
Even a child will not be said to as yet know its numbers if its counting-like actions begin at times with one, and at times with three, and so on. Counting isn't just any kind of repetition of familiar sounds together with gestures of pointing.
P. But surely miscounting is counting. Just as invalidly inferring is inferring.
C. A child can miscount only if it has mastered the technique of counting. The very acts that with us are miscountings would not be that if they constituted the normal or typical performance of a child.
But we are drifting. Let's review where we're at.
We've moved from talking about axioms and their instances to the topic of the general and the particular in arithmetic. The question we are now considering could be put as follows: Is knowledge of the laws of number somehow implicit in our knowledge of the particular numbers and their sums and products?
I am inclined to think this is so. You are inclined to doubt it. You feel that knowledge of the numbers - which are which, and what the various sums and products are - comes both first and quite independently of knowledge of laws. I feel that knowledge of laws is implicit is our knowledge of the numbers.
I feel that laws are implicit even in an arithmetic entirely lacking the means for putting laws into words. This is connected with the importance I attach to the systematicity of arithmetic. But you feel that knowledge of which numbers are which, and what the sums and products are, are points of information which can be picked up one by one, in a non-systematic way, and that a system emerges only as we see certain patterns in the particular number facts we arrive at one by one.
Do I get our two perspectives right when I describe them in this way? What do you think?
P. Yes. I think so. And now it strikes me that you have perhaps moved away from your earlier view that arithmetical knowledge could be based on knowledge of axioms and deduction. For now you see the axioms as there in arithmetic even when the arithmetic in question cannot so much as formulate them as axioms. For what you say of laws holds for axioms, since axioms are laws.
C. Yes. It no longer is clear to me that mathematical knowledge comes down to knowledge of laws plus deduction from laws. The knowledge of the laws is, as it were, there from the start as knowing how to count, to add, and to multiply.
I would now prefer to say that talk in terms of axioms and deduction no more than alludes to what is going on in obtaining arithmetical knowledge - or represents that knowledge in a certain way. It puts things, we might say, in a «deductive style». And that might even be highly misleading presentation of that knowledge.
P. Well - I'm certainly pleased that you've finally become quite clear in your statements!
C. Yes. I too thought I was doing a lot better.
So here we are. Confused. Not a bad starting point for philosophy.
There are the particularities of the numbers, and the laws. The real issue, as I now see things, has nothing in particular to do with axioms. They are of interest only when one wants, as it were, to logically sum up a body of knowledge. Instead, it is the relation of the laws to the particularities that concerns us, and how law and particularity connect up with knowledge.
Or I might put it this way: We are wondering about the relation between non-quantificational and quantificational arithmetic.
So we need to ask ourselves: What are the roles of generality in arithmetic?
And here three things immediately present themselves: First, there is the generality present in our methods of calculating particularities. Second, there is the generality that enters into the formation of new mathematical concepts, and, third, there is the generality we arrive at through proof.
University of Nebraska-Lincoln
<csayward at unlserve.unl.edu>