Sean Carroll talks a lot of sense on the nature of science, and in a recent post gave a definition of “science” that focuses on the methods and attitudes of science. He wrote:

Science consists of the following three-part process:

1. Think of every possible way the world could be. Label each way an “hypothesis”.

2. Look at how the world actually is. Call what you see “data” (or “evidence”).

3. Where possible, choose the hypothesis that provides the best fit to the data.

It’s a good operational definition, separating a scientific attitude from a pseudo-scientific or religious attitude, although the process needs to be iterative, with the hypothesis of step 3 then being tested against new data; and I would add in Feynman’s maxim: try hard to see whether you are fooling yourself, remembering that yourself is the easiest person to fool.

Later in the article, however, I start disagreeing with Carroll about what is and isn’t science. He says:

Which is not to say that every worthwhile intellectual endeavor is a version of science in some way. Math and logic are not science, because they don’t involve steps 2 or 3. They are all about figuring out all possible ways that things could be, whether or not things actually are that way in our real world.

Hmm, is that really true? If it is true, why is it that maths and logic are such brilliant tools for describing our universe? And why do we have axioms that result in 1 + 1 = 2 (matching our universe) and not, say, 1 + 1 = 3?

There are two possible answers. First, it may be that self-consistent logical and mathematical systems must have the property 1 + 1 = 2, and that there is no possible system in which 1 + 1 = 3. The other possible answer is that we have adopted the axioms we have because they work in our universe.

The modern foundation of arithmetic is Peano’s axioms, but are these entirely arbitrary, or the only ones possible, or chosen because that’s how *our* universe works? If the last answer holds then maths and logic are indeed part of science under Carroll’s scheme.

As a matter of history our maths did derive empirically from our universe. The ancient Babylonians and Egyptians and Greeks looked to maths because of its utility, for counting, for calculating debts and interest payments, for timekeeping, for calculating distances, for navigation, for building, et cetera. The first discussions of π were empirical, resulting from drawing circles with a compass. The first discussions of right-angled triangles and Pythagoras’s theorem were all about the need for right angles for constructing buildings and similar tasks.

Early mathematicians developed these ideas because they worked, because they applied to the real world, and allowed us to predict the movements of the sun, moon and planets, and allowed us to navigate accurately. Even something as basic to maths as the counting numbers would have originated from, well, counting: one apple, two apples, three apples, four. It’s no accident that the mathematicians’ name for these numbers is the “natural” numbers.

From there mathematics was distilled into axioms, and then mathematicians set about exploring the consequences of the axioms. Yet this whole enterprise is still empirical in origin. Reasoning from empirically derived axioms using empirically derived logic is still empirical, it is just teasing out the consequences of empirical observations, and thus is still science.

It is a remarkable fact that purely theoretical reasoning from axioms is often, later on, found to have real-world correspondences (for example the curved non-Euclidean geometries developed abstractly by mathematicians were later found to describe gravity in General Relativity). This is surely because the fundamental axioms have a deep correspondence to our universe — which is not at all surprising if, at root, our most basic axioms in mathematics and logic are distillates of our empirical experience.

Are there alternative logical systems that do not use any empirically derived axioms, that build instead on, say, 1 + 1 = 3, and that simply don’t work when applied to our universe? If there are they are not the systems usually studied by mathematicians.

Sean Carroll then places a second area outside science:

On the other hand, things like aesthetics and morality aren’t science either, because they require an additional ingredient — a way to pass judgment, to say that something is beautiful/ugly or right/wrong.

In my opinion this is nearly right, but not fully right, because it suggests a wrong way of thinking about the issue. It suggests that something could indeed be abstractly or objectively beautiful or ugly or right or wrong. And if that were so then Carroll is correct that one would need a method to establish such a declaration.

But to even imply this is to think about the issue wrongly. Attributes of beauty or ugliness or rightness and wrongness are *opinions*, and it makes no sense to divorce such an attribute from the person doing the opining.

We don’t need to establish abstract standards of beauty or morals, we simply need to recognise these things as opinions. Science cannot establish these abstract standards because the very idea is, literally, nonsensical. It literally has no meaning — and science cannot give answers to ill-posed and meaningless questions, other than to declare the question ill-posed.

Thus there is nothing about morals and aesthetic judgements that is outwith the remit of science. Any sensibly posed question about them can (in principle) be answered by science, because such questions are about our *opinions* (“Does George like chocolate?”, “Is van Gogh considered to be a great painter?”, “Do people consider adultery to be wrong?”, “Is it acceptable manners to be 30 mins late?”; all of these are empirical questions).

Thus, the statement that Carroll considers, namely “killing babies is wrong”, is really a shorthand for “humans generally are of the opinion that killing babies is wrong”, which is equivalent to “humans generally have an emotional dislike of babies being killed and want it not to happen”.

Carroll, correctly, asserts that there is nothing *objective* about the statement “killing babies is wrong”, but he also asserts that it is not a *scientific* statement. Well **it is a scientific statement in its properly stated form!** It is only “not scientific” in its shortened, abstract form — if taken as standalone — because in that form it isn’t anything; in that form it has no meaning.

Anyone disagreeing is invited to explain what it actually means, without referring to any human opinion or feeling and without merely making a question-begging re-phrasing using near synonyms. Saying that we “ought” not do it, or would “deserve” to be punished, or similar, does not answer the question unless one also independently defines those words.

**Related posts:**

Six reasons why objective morality is nonsense

Science can answer morality questions

What does “science” in “scientism” mean?

Neil RickertHow about: 1 + 1 = 0 (used in mod 2 arithmetic)?

CoelPost authorThat’s saying the same thing, just in a different notation. 😉

Yonatan FishmanRegarding the sentence:

“It is a remarkable fact that purely theoretical reasoning from axioms is often, later on, found to have real-world correspondences (for example the curved non-Euclidean geometries developed abstractly by mathematicians were later found to describe gravity in General Relativity). This is surely because the fundamental axioms have a deep correspondence to our universe — which is not at all surprising if, at root, our most basic axioms in mathematics and logic are distillates of our empirical experience.”

But doesn’t this contradict what you said earlier- namely, that the axioms are ultimately derived from our experience? Clearly, the distinguishing axioms of non-Euclidean geometries (specifically, negations of the parallel postulate) were not distilled from our immediate experience in the world.

Therefore, in my view, the best way to understand the nature of mathematics and its relation to the physical world is by considering that mathematics actually consists of two inter-related ‘activities’: ‘game’ or ‘formal’ mathematics and ’empirical’ or ‘applied’ mathematics. In ‘game’ mathematics we are free to choose and experiment with whatever axioms and rules of inference we please, whether or not they correspond with anything in the world. Theorems derived in such axiomatic systems are ‘necessarily true’ only insofar as they logically follow from the chosen set of axioms via the stipulated rules of inference. In ’empirical’ mathematics the axioms and rules of inference are inspired by our experience in the world (hence, explaining why math works in describing nature). While the axiomatic systems of ‘game’ mathematics need not correspond to anything in the physical world, sometimes, fortuitously, they will (as exemplified by the application of non-Euclidean geometry to relativity). For a chapter-length defense of this view, see Simon Altmann’s book, Is Nature Supernatural? (Prometheus books). For a similar view, see Morris Kline’s excellent book, Mathematics and the Physical World.

CoelPost authorHi Yonatan,

I think there is sense in that view, and indeed I am drafting a blog post which says something similar to that.

Yonatan FishmanAlso, Carroll’s #3 “Where possible, choose the hypothesis that provides the best fit to the data” is inadequate. There is more to the strength of a theory than merely its ability to fit the data. After all, any outrageous conspiracy theory could be invented to fit the data perfectly (just as a potentially infinite number of curves can be drawn to exactly fit a given set of data points). Thus, what is missing from Carroll’s account is parsimony.