Related Sequence posts: Math is Subjunctively Objective, How to Convince Me That 2+2=3
Discussions whether mathematical theorems can be possibly disproved by observations have been relatively frequent on LessWrong. The last instance which motivated me to write this post was the discussion here. In fact these discussions are closely related to philosophical disputes about contingent and necessary truths. Many standard philosophical disputes are considered solved on LessWrong and there is usually some reference post which dissolves the issue. I don't know of any post that conclusively summarises this problem, although it doesn't seem particularly controversial.
To be concrete, let's take a single statement of arithmetic, say "5324 + 2326 = 7650". Most people will gladly agree that the statement is true. No matter how proficient the debaters are in intellectual sophistry (with the possible exception of postmodern philosophers), they will not dispute that 5324 + 2326 indeed equals to 7650. But the agreement is lost when it comes to the meaning of the discussed simple sentence. What does it refer to? Is it a necessary truth, or can it be empirically tested?
One opinion states that statements about addition refer to counting apples, pigs, megabytes of data or whatever stuff one needs to count. Sets of these objects, either material or abstract, are the referents of those statements. If I take two groups of apples which happen to contain 5324 and 2326 items and put them together, the compound group will consist of 7650 apples. It is not self-evident (as any illiterate tribesman from New Guinea would confirm) that the result will be 7650 and not, for example, 7494. Therefore each putting of apples together counts as a test of the respective proposition about arithmetic. Of course, the number of apples in the compound group must be determined by a method different from counting the two subgroups separately and then adding the numbers on the paper — that would defeat the idea of testing. Also, some people may say that "5324 + 2326" is a definition of "7650"; not that we regularly encounter such opinions when speaking about numbers of this magnitude, but that "1 + 1" is a definition of "2" I have heard countless times. So we have to be careful to create a dictionary which translates "1" as "S0" and "2" as "SS0" and "7650" as something which I am not going to write here, and describe counting of apples as adding one S for each apple in the bag. After that we may go through the ordeal of converting our bag of apples to a horribly long string of S's and finally look up our product in the dictionary - just to see whether the corresponding translation really reads "7650". And if done in this torturously lengthy way, I assume there would be at least a tiny amount of pleasant surprise if we really found "7650" and not "157", and not only because the possibility of an "experimental error". So it seems to be a legitimate empirical test.
Holders of the contrary opinion would certainly not deny that such tests can be arranged (they may dispute the "surprise", though). But their argument is: Even if we conducted the described experiment with apples, and repeatedly found that the result was 157 instead of 7650 (and suppose that the possible errors in counting or translation from "5324" to "SSS...S0" were ruled out), that has no bearing on the truth value of "5324 + 2326 = 7650" as a statement of arithmetic. It is imaginable that physical addition of apples followed some different rules, such that putting 5324 objects together with 2326 objects always yielded a set of 157 objects — but that doesn't mean that "5324 + 2326 = 157". There would be an isomorphism, in such a hypothetical world, which maps that string to a true statement about apples, nevertheless there is no way how to make it a statement about arithmetic. It would be better to invent a new symbol for the abstract operation which emulates physical addition in that hypothetical world, or even better whole set of new symbols for all digits, to avoid confusion. One may rather say "%#@$ ¯ @#@^ = !%&" instead of "5324 + 2326 = 157". The former is a statement of a certain formal system X which models the modified apple addition and as such it is true, while the latter is false. No matter what apples do in that world, within arithmetic we can still formally prove that 5324 + 2326 is 7650, and nothing else. Even if the inhabitants of our strange hypothetical world called their formal counting system "arithmetic" instead of "X" and the existence of real arithmetic had never occured to them — even such a fact cannot change the universal truth that 5324 + 2326 = 7650.
Although there is hardly any disagreement about expected anticipations, the debates on this question seldom appear conclusive. The apparent disagreement is almost certainly caused by different interpretation of something. It is perhaps not much different from the iconic sound definition dispute or the disputes about morality. In contrast to the sound definition case, where the disagreement is about what a single word "sound" refers to, here the source of misunderstanding is more difficult to locate. On the first sight it may appear that the meaning of "arithmetic" is disputed. However more probably it is the phrase "5324 + 2326 = 7650" with all other statements of arithmetic which is interpreted in several distinct ways. Let's be more specific in what the proposition can mean:
- If I take 5324 objects and add another 2326 objects, I get 7650 objects. It holds for a broad range of object types and all reasonable senses of "adding together", therefore it is sensible to express the fact as a general abstract relation between numbers. By the way, we can create a formal system which allows us to deduce similar true propositions, and we call it "arithmetic".
- The string "5324 + 2326 = 7650" is a theorem in a formal system given by the following axioms and rules: (Here should stand the axioms and rules.) By the way, we call the system "arithmetic", and it happens to be a good model of counting objects.
(There might be a third interpretation, along the lines of the second one, but with less apparent arbitrariness of arithmetic: "any intelligence necessarily includes representation of a formal system isomorphic to arithmetic, independently of the properties of the external world". I didn't include that to the list, because it is either a very narrow constraint on the definition of intelligence or almost certainly false.)
Because arithmetic actually works (as far as we know) as a model of counting, the two interpretations are equally good and for all practical purposes indistinguishable. It is no surprise that our intuitions can't reliably distinguish between practically equivalent interpretations. Rather, two intuitions come into conflict. The first one tells us that arithmetic isn't arbitrary at all, and thus the second interpretation must be false. The second intuition is based on the self-consistence of mathematics: mathematics has its own ways how to decide between truth and falsity, and those ways never defer to the external world; therefore the first interpretation must be false. But once the meaning is spelled out in sufficient detail, the apparent conflict should disappear.
It seems that you prefer the second interpretation of mathematical statements. But the first one, that which refers to physical world, isn't completely unattractive either.
For example I can use multiplication only for calculating areas of rectangles; if so, I would probably hold that "5x3" means "area of a rectangle whose sides measure 5 and 3", and "there is a real number which multiplied by itself equals two" means "there is a square of area 2". I would say that 5 times 3 equals 15 if and only if it was true for all real rectangles which I have met, and after seeing a counter-example I would abandon the belief that "5x3=15".
Or I can mean "if I add together five groups of three apples each, I would find fifteen objects". If this was my understanding of what the proposition means, a counter-example consisting of a 5x3 rectangle with area of 27 wouldn't persuade me to abandon the abstract belief, because multiplication is not inherently about rectangles.
In the real world people are familiar with both uses of multiplication and many others, so any counter-example in one area is likely to be perceived as evidence that multiplication isn't a good model of that process, rather than that we have to update our understanding of multiplication.
Math is exact in the sense that once the rules of inference are given there is no freedom but to follow them, and unobjectionable in the sense that it is futile to dispute the axioms. Any axiomatic system is like that. But most mathematical models we actually used have one advantage over that: they are not arbitrary, but rather designed to be useful to describe plethora of different real-world situations. Removing a single application of multiplication doesn't shatter the abstract truth, but if you consecutively realised that multiplication does capture neither calculating areas nor putting groups of objects together nor any other physical process, what content would remain in propositions like "5x3=15"? They would be meaningless strings produced by an arbitrary prescription.
As a quick aside, I think these two interpretations are actually the same thing in disguise. Areas as measurements have units attached to the numbers. Specifically, the units are squares whose sides measure one "unit length". So when you're looking at a rectangle that measures 5x3, you're noting that there are five groups of three squares (or three groups of five squares, depending on how you want to interpret the roles of the factors). Otherwise it's hard to see why the area would be a result of multiplying the lengths of the sides.
I think perhaps a better example would be the difference between partitive and quotative division. Partitive ("equal-sharing") says "I have X things to divide equally between N groups. How many things does each group get?" Quotative ("measurement" or "repeated subtraction") says "I have X things, and I want to make sure that each group gets N of those things. How many groups will there be?" This is the source of not a small amount of confusion for children who are taught only the partitive interpretation and are given a jumble of partitive and quotative division word problems. It's not immediately obvious why these two different ideas would result in the same numerical computation; it's actually a result of the commutativity of multiplication and the fact that division is inverse multiplication. So there's a deep structure here that's invisible even to participants that still guides their activities and understanding.
I agree that axiomatic systems are like that, but I don't think the essence of math is axiomatic. That's one method by which people explore mathematics. But there are others, and they dominate at least as much as the axiomatic method.
For instance, Walter Rudin's book Real and Complex Analysis goes through a marvelously clean and well-organized axiomatic-style exposé of measure theory and Lebesgue integration. But I remember struggling with several of my classmates while going through that class trying to make sense of what is "really going on". If math were just axiomatic, there wouldn't be anything left to ask once we had recognized that the proofs really do prove the theorems in question. But there's still a sense of there being something left to understand, and it certainly seems to go beyond matters of classification.
What finally made it all "click" for me was Henri Lebesgue's own description of his integral. I can't seem to find the original quote, but in short he provided an analogy of being a shopkeeper counting your revenue at the end of the day. One way, akin to the Riemann integral, is to count the money in the order in which it was received and add it up as you go. The second, akin to Lebesgue integration, is to sort the money by value - $1 bills, $5 bills, etc. - and then count how many are in each pile (i.e. the measure of the piles). This suddenly made everything we were doing make tremendously more sense to me; for instance, I could see how the proofs were conceived, even though my insight didn't actually change anything about how I perceived the axiomatic logic of the proofs.
The fact that some people saw this without Lebesgue's analogy is beside the point. The point is that there's an extra something that seems to need to be added in order to feel like the material is understood.
I'm going to some lengths to point this out because the idea of math as perfect and axiomatic just isn't the mathematics that humans practice or know. It can look that way, but the truth seems to be more complicated than that.