I’m having issues with the comments section at Philosophy Bro, so I’ll post this here instead. In a discussion of empiricism and rationalism, the author writes:

You are *definitely *over-committing yourself if you mean that *everything *we can know is mediated through the senses – I mean, mathematics? Logical tautologies?

As with my earlier discussion of science, the key missing idea here is that of the map/territory relation. If scientists and philosophers would realize that they’re building models/maps to explain the world, and not defining rules that govern how the world/territory works, our understanding of these issues would improve drastically.

Frankly, I find it preposterous to suggest that mathematics can be known without relying on sensory perception. Show me a way to teach a child addition that doesn’t rely on sensory perception–no blocks, no pictures, no counting on your fingers. Then, and only then, will I grant that mathematics can be intuited without sensory perception. Yes, armchair philosophers can understand math without reference to specific measurements. But only because they’ve previously perceived an enormous number of examples of mathematical theory working.

Based on observation, we create a model; we test it; it works; over time we accept it as very strong theory. That’s our inductive process, for science, for math, for logic. After we have our model, we extrapolate, interpolate, and infer–deductive processes. So we can figure out that 9823+2349=12172 without measuring. But only because we’ve previously seen so many other applications of our addition theory work, without ever seeing one fail. We *trust* that we don’t need to test it, based on the enormity of evidence supporting the model. But our trust doesn’t make the theory *true*.

(Editorial note: Philosophy Bro writes casually about interesting issues in philosophy. In a previous post about Free Will, the author used a word I found offensive, and I said so in the comments. The word remains in that post, and as such I won’t link to it. However, I like the blog, and am wiling to write off one isolated case of bad judgment in linking to the site.)

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## 4 Comments

Sorry, mathematics can be known without sensory data. The Pythagorean Theorem is a necessary consequence of certain assumptions about planar geometry. A theorem-proving computer program will derive it without any sensory input or intelligence whatsoever. If we should ever meet intelligent aliens, we can be confident that they will agree with us on its truth no matter how different our senses or psychology.

The Pythagorean Theorem is also not a model of anything. Not only do you not need to observe any real triangles to prove it, you cannot do so – Euclid’s assumptions are not true in the world where we live.

It appears, like most mathematics, to be a truth independent of the human mind – something we discovered rather than invented.

SC,

Thanks for chiming in. No need to apologize–I’ve labeled myself as a contrarian, so I’m pretty comfortable when people disagree with my analysis.

As you say, Euclidean geometry is not a perfect representation of the universe. Rather, it’s a model, which when applied to the real world, in fields like engineering and astronomy, provides valuable insight. The Pythagorean Theorem can be derived/proven based on Euclid’s assumptions, but this doesn’t make it “true”; it just makes it a derivative of the Euclidean model, which is a very strong theory.

Cartesian geometry, which relies on a different set of assumptions, can also be used to derive the Pythagorean Theorem, but this doesn’t make it true either; it just means it’s a consequence of more than one model. (See my earlier post on the utility of having more than one map)

The process Euclid used to develop the geometry is no different than what I describe. He didn’t invent from thin air; rather, he based his theory on observations of shapes and figures. After observing many shapes that are very close to straight lines and circles, he imagined a model that clearly defines straight lines and circles. He tested his theory by making predictions–for instance by creating right-angle triangles with lengths 3,4,5 or 5,12,13. These predictions held, and no counter-examples were found; over time, the model was accepted to be very strong. That’s induction. It yields very strong theories, but not truths.

I agree with what you say about induction. I’d also agree that any branch of human knowledge that uses induction is going to require evidence from the human senses. Engineering & architecture require empiricism.

But as Phil Bro says, that’s not all of knowledge. Much of mathematics, including Euclidean geometry, is deductive. It logically reaches conclusions from premises. We, in fact, don’t require induction over a large set of triangles to prove the Pythagorean Theorem. That’s why a completely mechanical computer program can prove it. It’s a truth that does not depend on human senses and does not require the human mind.

A mechanical computer can also “prove” that I have no reflection if you input the assumption that I’m a vampire. But any actual intelligence would want to test that conclusion. When it determines I do indeed have a reflection, it would discredit the theory that I’m a vampire.

At some point, you need to test Euclidean geometry–either its assumptions or its conclusions, and preferably both, against real-world phenomena. Without testing, you don’t have any knowledge, just conjecture. And a claim derived from empirically validated assumptions is entirely dependent on inductive reasoning.

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