Math is Subjunctively Objective

Followup to: Probability is Subjectively Objective, Can Counterfactuals Be True?

I am quite confident that the statement 2 + 3 = 5 is true; I am far less confident of what it means for a mathematical statement to be true.

In “The Simple Truth” I defined a pebble-and-bucket system for tracking sheep, and defined a condition for whether a bucket’s pebble level is “true” in terms of the sheep. The bucket is the belief, the sheep are the reality. I believe 2 + 3 = 5. Not just that two sheep plus three sheep equal five sheep, but that 2 + 3 = 5. That is my belief, but where is the reality?

So now the one comes to me and says: “Yes, two sheep plus three sheep equals five sheep, and two stars plus three stars equals five stars. I won’t deny that. But this notion that 2 + 3 = 5, exists only in your imagination, and is purely subjective.

So I say: Excuse me, what?

And the one says: “Well, I know what it means to observe two sheep and three sheep leave the fold, and five sheep come back. I know what it means to press ‘2’ and ‘+’ and ‘3’ on a calculator, and see the screen flash ‘5’. I even know what it means to ask someone ‘What is two plus three?’ and hear them say ‘Five.’ But you insist that there is some fact beyond this. You insist that 2 + 3 = 5.”

Well, it kinda is.

“Perhaps you just mean that when you mentally visualize adding two dots and three dots, you end up visualizing five dots. Perhaps this is the content of what you mean by saying, 2 + 3 = 5. I have no trouble with that, for brains are as real as sheep.”

No, for it seems to me that 2 + 3 equaled 5 before there were any humans around to do addition. When humans showed up on the scene, they did not make 2 + 3 equal 5 by virtue of thinking it. Rather, they thought that ‘2 + 3 = 5’ because 2 + 3 did in fact equal 5.

“Prove it.”

I’d love to, but I’m busy; I’ve got to, um, eat a salad.

“The reason you believe that 2 + 3 = 5, is your mental visualization of two dots plus three dots yielding five dots. Does this not imply that this physical event in your physical brain is the meaning of the statement ‘2 + 3 = 5’?”

But I honestly don’t think that is what I mean. Suppose that by an amazing cosmic coincidence, a flurry of neutrinos struck my neurons, causing me to imagine two dots colliding with three dots and visualize six dots. I would then say, ‘2 + 3 = 6’. But this wouldn’t mean that 2 + 3 actually had become equal to 6. Now, if what I mean by ‘2 + 3’ consists entirely of what my mere physical brain merely happens to output, then a neutrino could make 2 + 3 = 6. But you can’t change arithmetic by tampering with a calculator.

“Aha! I have you now!”

Is that so?

“Yes, you’ve given your whole game away!”

Do tell.

“You visualize a subjunctive world, a counterfactual, where your brain is struck by neutrinos, and says, ‘2 + 3 = 6’. So you know that in this case, your future self will say that ‘2 + 3 = 6’. But then you add up dots in your own, current brain, and your current self gets five dots. So you say: ‘Even if I believed “2 + 3 = 6″, then 2 + 3 would still equal 5.’ You say: ‘2 + 3 = 5 regardless of what anyone thinks of it.’ So your current brain, computing the same question while it imagines being different but is not actually different, finds that the answer seems to be the same. Thus your brain creates the illusion of an additional reality that exists outside it, independent of any brain.”

Now hold on! You’ve explained my belief that 2 + 3 = 5 regardless of what anyone thinks, but that’s not the same as explaining away my belief. Since 2 + 3 = 5 does not, in fact, depend on what any human being thinks of it, therefore it is right and proper that when I imagine counterfactual worlds in which people (including myself) think ‘2 + 3 = 6’, and I ask what 2 + 3 actually equals in this counterfactual world, it still comes out as 5.

“Don’t you see, that’s just like trying to visualize motion stopping everywhere in the universe, by imagining yourself as an observer outside the universe who experiences time passing while nothing moves. But really there is no time without motion.”

I see the analogy, but I’m not sure it’s a deep analogy. Not everything you can imagine seeing, doesn’t exist. It seems to me that a brain can easily compute quantities that don’t depend on the brain.

What? Of course everything that the brain computes depends on the brain! Everything that the brain computes, is computed inside the brain!”

That’s not what I mean! I just mean that the brain can perform computations that refer to quantities outside the brain. You can set up a question, like ‘How many sheep are in the field?’, that isn’t about any particular person’s brain, and whose actual answer doesn’t depend on any particular person’s brain. And then a brain can faithfully compute that answer.

If I count two sheep and three sheep returning from the field, and Autrey’s brain gets hit by neutrinos so that Autrey thinks there are six sheep in the fold, then that’s not going to cause there to be six sheep in the fold—right? The whole question here is just not about what Autrey thinks, it’s about how many sheep are in the fold.

Why should I care what my subjunctive future self thinks is the sum of 2 + 3, any more than I care what Autrey thinks is the sum of 2 + 3, when it comes to asking what is really the sum of 2 + 3?

“Okay… I’ll take another tack. Suppose you’re a psychiatrist, right? And you’re an expert witness in court cases—basically a hired gun, but you try to deceive yourself about it. Now wouldn’t it be a bit suspicious, to find yourself saying: ‘Well, the only reason that I in fact believe that the defendant is insane, is because I was paid to be an expert psychiatric witness for the defense. And if I had been paid to witness for the prosecution, I undoubtedly would have come to the conclusion that the defendant is sane. But my belief that the defendant is insane, is perfectly justified; it is justified by my observation that the defendant used his own blood to paint an Elder Sign on the wall of his jail cell.’”

Yes, that does sound suspicious, but I don’t see the point.

“My point is that the physical cause of your belief that 2 + 3 = 5, is the physical event of your brain visualizing two dots and three dots and coming up with five dots. If your brain came up six dots, due to a neutrino storm or whatever, you’d think ‘2 + 3 = 6’. How can you possibly say that your belief means anything other than the number of dots your brain came up with?”

Now hold on just a second. Let’s say that the psychiatrist is paid by the judge, and when he’s paid by the judge, he renders an honest and neutral evaluation, and his evaluation is that the defendant is sane, just played a bit too much Mythos. So it is true to say that if the psychiatrist had been paid by the defense, then the psychiatrist would have found the defendant to be insane. But that doesn’t mean that when the psychiatrist is paid by the judge, you should dismiss his evaluation as telling you nothing more than ‘the psychiatrist was paid by the judge’. On those occasions where the psychiatrist is paid by the judge, his opinion varies with the defendant, and conveys real evidence about the defendant.

“Okay, so now what’s your point?”

That when my brain is not being hit by a neutrino storm, it yields honest and informative evidence that 2 + 3 = 5.

“And if your brain was hit by a neutrino storm, you’d be saying, ‘2 + 3 = 6 regardless of what anyone thinks of it’. Which shows how reliable that line of reasoning is.”

I’m not claiming that my saying ‘2 + 3 = 5 no matter what anyone thinks’ represents stronger numerical evidence than my saying ‘2 + 3 = 5’. My saying the former just tells you something extra about my epistemology, not numbers.

“And you don’t think your epistemology is, oh, a little… incoherent?”

No! I think it is perfectly coherent to simultaneously hold all of the following:

  • 2 + 3 = 5.

  • If neutrinos make me believe “2 + 3 = 6”, then 2 + 3 = 5.

  • If neutrinos make me believe “2 + 3 = 6”, then I will say “2 + 3 = 6″.

  • If neutrinos make me believe that “2 + 3 = 6”, then I will thereafter assert that “If neutrinos make me believe ‘2 + 3 = 5’, then 2 + 3 = 6″.

  • The cause of my thinking that “2 + 3 = 5 independently of what anyone thinks” is that my current mind, when it subjunctively recomputes the value of 2 + 3 under the assumption that my imagined self is hit by neutrinos, does not see the imagined self’s beliefs as changing the dots, and my current brain just visualizes two dots plus three dots, as before, so that the imagination of my current brain shows the same result.

  • If I were actually hit by neutrinos, my brain would compute a different result, and I would assert “2 + 3 = 6 independently of what anyone thinks.”

  • 2 + 3 = 5 independently of what anyone thinks.

  • Since 2 + 3 will in fact go on equaling 5 regardless of what I imagine about it or how my brain visualizes cases where my future self has different beliefs, it’s a good thing that my imagination doesn’t visualize the result as depending on my beliefs.

“Now that’s just crazy talk!”

No, you’re the crazy one! You’re collapsing your levels; you think that just because my brain asks a question, it should start mixing up queries about the state of my brain into the question. Not every question my brain asks is about my brain!

Just because something is computed in my brain, doesn’t mean that my computation has to depend on my brain’s representation of my brain. It certainly doesn’t mean that the actual quantity depends on my brain! It’s my brain that computes my beliefs about gravity, and if neutrinos hit me I will come to a different conclusion; but that doesn’t mean that I can think different and fly. And I don’t think I can think different and fly, either!

I am not a calculator who, when someone presses my “2” and “+” and “3” buttons, computes, “What do I output when someone presses 2 + 3?” I am a calculator who computes “What is 2 + 3?” The former is a circular question that can consistently return any answer—which makes it not very helpful.

Shouldn’t we expect non-circular questions to be the normal case? The brain evolved to guess at the state of the environment, not guess at ‘what the brain will think is the state of the environment’. Even when the brain models itself, it is trying to know itself, not trying to know what it will think about itself.

Judgments that depend on our representations of anyone’s state of mind, like “It’s okay to kiss someone only if they want to be kissed”, are the exception rather than the rule.

Most quantities we bother to think about at all, will appear to be ‘the same regardless of what anyone thinks of them’. When we imagine thinking differently about the quantity, we will imagine the quantity coming out the same; it will feel “subjunctively objective”.

And there’s nothing wrong with that! If something appears to be the same regardless of what anyone thinks, then maybe that’s because it actually is the same regardless of what anyone thinks.

Even if you explain that the quantity appears to stay the same in my imagination, merely because my current brain computes it the same way—well, how else would I imagine something, except with my current brain? Should I imagine it using a rock?

“Okay, so it’s possible for something that appears thought-independent, to actually be thought-independent. But why do you think that 2 + 3 = 5, in particular, has some kind of existence independently of the dots you imagine?”

Because two sheep plus three sheep equals five sheep, and this appears to be true in every mountain and every island, every swamp and every plain and every forest.

And moreover, it is also true of two rocks plus three rocks.

And further, when I press buttons upon a calculator and activate a network of transistors, it successfully predicts how many sheep or rocks I will find.

Since all these quantities, correlate with each other and successfully predict each other, surely they must have something like a common cause, a similarity that factors out? Something that is true beyond and before the concrete observations? Something that the concrete observations hold in common? And this commonality is then also the sponsor of my answer, ‘five’, that I find in my own brain.

“But my dear sir, if the fact of 2 + 3 = 5 exists somewhere outside your brain… then where is it?”

Damned if I know.

Part of The Metaethics Sequence

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