21 Logic, Math, and Science
We should begin by situating scientific knowledge among other types of knowledge. As we have seen, some known propositions are contingent, meaning that they express facts that easily could have been otherwise. As it happens, for example, some lucky person in Canyon County won the Idaho Powerball jackpot in August 2017. That could have been otherwise—someone else could have won, or the prize might have gone unclaimed for another month. It snowed last Thursday (again), so weather patterns might have shifted slightly, bringing the snow sooner or later or not at all. Of course, for these events to have been otherwise, the particular causes would have had to have been different, and for those particular causes to have been different, their causes would have had to have been different, and so on. But none of these changes seem impossible. Each change is “thinkable” or imaginable on its own.
Other known propositions are necessary, meaning that they really could not be otherwise. So in geometry, for example, a cube contained within a sphere has less volume than the sphere. It’s hard to get around that fact—there is no way it could be otherwise, given the meanings of the terms we are using. In these causes, it is not a matter of re-engineering causes to bring about different effects. The changes themselves are unthinkable or unimaginable.
(Someone might wonder if this cube/sphere proposition might be considered contingent since it is, after all, contingent that the words “cube” and “sphere” mean what they do in English. Clearly, those words might have meant different things. It is a good question. One reason philosophers like to talk about propositions is that a proposition is supposed to be the meaning of what is said in whatever language. So, yes, the sentence, “The cube is in the sphere” might have meant many different things or nothing at all; but the proposition that the cube is in the sphere means precisely one thing: the thing that is also meant when we say, “el cubo está en la esfera” or “a kocka a gömbben van” or “tha an ciùb anns an raon” which all mean “the cube is in the sphere.”)
The known propositions of logic and mathematics are necessary. How do we know they are necessary? Is it simply a matter of what we can or cannot imagine? This is a very good question. A first answer might be that we know these propositions are necessary because if we deny them, then we can derive a contradiction from them. So, for example, five plus three equals eight is a true and necessary proposition. If we try to deny it, we end up in the following sort of trouble:
5 + 3 ≠ 8 (suppose)
(xxxxx) + (xxx) ≠ (xxxxxxxx) (by definitions of “5”, “3”, and “8”)
(xxxxxxxx) ≠ (xxxxxxxx) (by definition of “+”)
8 ≠ 8 (by definition of “8”)
ABSURD! (by definition of “≠”)
So we might say that necessary propositions are those whose denials entail contradictions or results that are false in virtue of the meanings of the terms. Perhaps this is a good enough answer. But some philosophers—notably W. V. Quine whom we encountered in the last chapter—have wondered whether “meanings of terms” are fixed in such precise ways as to allow for a clear distinction between necessary and contingent truths. Don’t we learn the meanings of terms in rather informal circumstances, which allow for quite a lot of slippage and unclarity and vagueness? So, for example, what about the claim that Catholic priests are male? Is that true in virtue of the meanings of the terms, or is it a contingent truth based on decisions made by a particular tradition? Is it obviously “more necessarily” true than the claim that some dogs have tails? We might wonder whether there really are hard-edged “meanings” of terms that allow us to definitively determine whether a given claim is necessarily true or contingently true. This objection is generally known as “Quine’s criticism of the analytic/synthetic distinction” and it is an interesting and important discussion to study, but it is a bit beyond our reach in this introduction.
So, having mentioned that objection, I will now set it aside, and continue as if we have some good way of distinguishing necessary truths from contingent truths. Logical truths (such as “P ⇔ P” or “if P ⇒ Q and Q ⇒ R, then P ⇒ R” or “if P v Q and ~Q, then P”) and mathematical truths (including all those in arithmetic, geometry, algebra, calculus, etc.) are necessary truths. We know they are necessary truths because if we try to deny them, we will find that we can derive claims that are false in virtue of the meanings of the terms. Other particular facts about the world such as what happened here or there, how long some particular rhino’s horn is, or who stole the cookies from the cookie jar, and so on, are contingent. We know they are contingent because if we deny them, we will find that we can derive claims that, in fact, are false but not false in virtue of the meanings of the terms. Denying that Slim Jim won the Idaho lottery, for instance, might make it harder for us to explain how he was able to afford a shiny new Cadillac, but it will not lead us to derive claims that are false by virtue of the meanings of “Cadillac,” “lottery,” or “Idaho.”
Now what about known propositions of science? In particular, what about the known propositions we identify as laws of nature? These known propositions seem to be somewhere between necessary and contingent. We can deny them without running into contradictions about meanings of terms. So, for example, suppose it is a law of nature that force equals mass times acceleration. Indeed, this was once thought to be a law of nature known as “Newton’s second law of motion.” It was thought to be a rock-solid truth, one perhaps that could not be otherwise. But since then, we have learned that this law not only could be otherwise, but it actually is otherwise since, to update it to Einstein’s theory of relativity, we need to complicate the equation a bit (i.e., taking into account how fast the observer is moving relative to the speed of light). Einstein made this advance upon Newton without running contrary to any of the meanings of the terms involved: “mass” still meant “mass,” but the relation to force and acceleration turned out to be a little different. So it is evidently possible to deny Newton’s second law of motion without entailing a contradiction. Moreover, it is possible to deny Einstein’s laws without entailing a contradiction. And, indeed, any of the known laws of nature can be denied without entailing a contradiction.
Does that make the laws of nature contingent? It does if we hold fast to the claim that contingent propositions are the ones that can be denied without entailing any contradiction. But at the same time, there is something about laws of nature that make them seem similar to necessary truths. Laws of nature are more fundamental—“closer to the core of reality,” so to speak—than contingent facts about particular things. When scientists discover basic laws of nature, they are getting at deep truths about reality, truths that could be different only if reality itself were different in some fundamental way. This depth of the truth of laws of nature makes them seem similar to truths in logic or mathematics, which also could be different only if reality itself were different in some really fundamental way.
Perhaps an example will make this idea clearer. Suppose we inflate a balloon until it bursts. We can imagine all sorts of ways to vary this exciting experiment: we could use thicker or thinner balloons, we could use different sorts of gases, we could inflate the balloon more or less quickly, we could do it on mountaintops or down in the valley, during the day or night, and so on. These are changes we can easily make. But suppose that instead of making any of these easy changes, we want to keep everything exactly the same but delay the bursting for an extra minute. That is to say, we want to use the same sort of balloon, the same gas, the same outside pressure, the same rate of inflation but just delay the bursting by a minute. To do this, we will have to change some natural fact that has to do with the strength of the balloon material. We will have to change a deep fact about the nature of the world and, specifically, about how much that sort of material can stretch before ripping. That’s really hard to do. In fact, for creatures like us, it is impossible, for humans cannot alter the laws of nature that govern the limits of materials.
Of course, in words, or conceptually, we can deny whatever law of nature that is involved in this experiment, and our denial will not entail any contradiction. But we cannot deny or change the law in fact. We cannot really make it false. There is a sort of necessity to the law of nature that simply is not found in the other particular circumstances, all of which we are able to change by using different balloon materials, a different gas, different altitudes, and so on.
So the denial of a law of nature is impossible, but for some reason other than that the denial of the law entails a contradiction. It would be interesting to continue to pursue this line of thought, but once again, this is a topic that takes us quickly into matters beyond the scope of this introduction. For our purposes, we might simply recognize three types of necessity: logical necessity, mathematical necessity, and natural necessity which is the sort of necessity pertaining to scientific laws of nature. The differences among these kinds of necessity are philosophically interesting, but we won’t pursue the topic here.
- I am sorry to say matters are more complicated than this. It has been proven that in mathematics there will always be true propositions that cannot be proven. If you deny them, you won't be able to derive a contradiction. For further explanation, look up "Gödel's Incompleteness Theorem." ↵