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.