I’ve been rereading a bunch of the stuff on Field in the last couple days (my exam is in about 14 hours – I hadn’t planned on publicizing this blog until afterwards, but I guess I ended up doing so early) and I’ve been struck by the thought that a lot of nominalist worries about the causal efficacy of mathematical objects could be put a bit more mildly.
In general I’m a believer in the Quine-Putnam argument, though I think that mathematics actually is dispensible, so it ends up being a nominalist argument. This means that if mathematics had turned out to be indispensible, then I’d have to believe in its entities and confront the epistemological worries. But I think they aren’t really a serious problem, because our theory can be confirmed as a whole. I do think there is a slight issue though, in that these numbers that enter into our physical theories aren’t located near the entities they’re correlated with, or connected to them in any obviously causal way. The appearance of abstract objects in a physical theory seems to me like it’s at least a minor point against the theory, because it’s basically a stronger version of the “action at a distance” that Einstein and the early modern scientists disliked so much.
Of course, if the best theory predicts action at a distance, then I guess we have to live with it, and if the best theory predicts action from nowhere then we have to live with that too. But it doesn’t mean I have to like it any more than I should have to like a theory that is essentially non-deterministic (like quantum mechanics) or has any of a number of other blemishes that I’d like to think an ideal scientific theory wouldn’t have.
Antimeta 
Good luck with the exam!
Thanks! I passed, and then we went out to the bar where John MacFarlane was fiddling. It was a good time.
Fantastic! Congrats.
Congrats!
Congratulations! Does this mean you can add “ABD” after your name now?
But let me ask why you dislike “action at a distance” or “non-determinism”? Is it not simply that we have a pre-existing theory (perhaps implicit, unarticulated, part of our folk-theory of physics, for example) that suggests such aspects are reasonable assumptions?
Cantor showed mathematicians that their pre-existing intuitions (about the infinite) may be false. That a theory supports (say) “action at distance” is not a reason to reject the theory, but a reason to examine our pre-existing intuitions.
I suppose that it seems to me that it’s part of the notion of what a scientific theory is. Until quantum mechanics came around, scientific theories were almost all deterministic, and the non-deterministic ones (like statistical mechanics and thermodynamics) always had places where the non-determinism could be filled in by determinism at some smaller scale.
At a recent talk I went to by psychologist Alison Gopnik, she pointed out that when four-year-olds are investigating a device to discover how it works, they don’t rest until they’ve found a deterministic causal chain where each part is attached to the next part. When the device is inherently randomized, or when two things that aren’t connected seem to causally interact, they keep searching for more “hidden variables”.
It’s certainly true that eventually we should allow intuitions to be revised by science, but we shouldn’t be too willing to immediately discard them either, or else we’d have no constraints on what would count as good science.
But, Kenny, surely the work of Cantor in Mathematics and of Einstein, Schrodinger & Heisenberg in Physics demonstrated that our intuitions are useless as guides as to what “counts as good science”. And how can anyone possibly have intuitions — either good or bad — about 11-dimensional space, the realms of String Theory and M-Theory?
Best to ignore our intuitions from the outset — At best, they will just get in the way, slowing us down; at worst, they will sidetrack us completely.
On the contrary – Einstein’s greatest work was based on intuition. While it conflicted some specific intuitions about what space was like, it wouldn’t have been possible if he hadn’t noted that the Newtonian/Maxwellian paradigm of the moment conflicted with intuitions that force fields should vary continuously and that the laws of physics shouldn’t discriminate between observation frames.
I’d like to see if the reasons some people opt for string theories rather than quantum theories is because they preserve certain intuitions about determinism or something else.
Kenny — You misread my post. I didn’t say that Einstein’s intuitions were wrong, I said that ours were. Great scientists *do* have great intuitions, but they (great scientists) are not usually the ones deciding what counts as good science. It is the rest of us who do this — the not-so-great who sit on research grant committees, who act as journal referees and editors, who decide promotions, along with the politicians, influenced by an educated public, deciding funding priorities.
Cantor too had exceptional intuitions, but they were certainly not shared by his contemporaries — he was driven to stop doing mathematics because of the opposition he faced from the leading mathematicians of his day, particularly Weierstrauss. He spend the second half of his life studying theology.
I think string theory is popular among theoretical physicists now because the math is elegant. That may or may not be a good reason for believing it to be a true depiction of reality. It certainly strikes me as insufficient justification for it to receive public funding. After all, the mathematics of theology could also be quite elegant.
And just to add a PS to my post: The physicist Frank Tipler has indeed shown that the mathematics of end-time Christian theology is indeed elegant!
Hi, my name is Cath and I’m a children’s writer, publishing in Australia. I don’t know whether you can help me but I really need someone to explain action at a distance to me as if I were a ten year old child. (You can assume, in your area of study, I have the mental capacity of a ten year old child.)
You can’t know how much I’d appreciate this.
Cath – I’m no expert on this stuff, but I’ll say what I can. For a more accurate representation, I would suggest looking on physics blogs, starting with Cosmic Variance (http://www.cosmicvariance.com/).
Anyway, the intuitive idea is that in order to affect something, you have to be close to it – you can push and pull things, but those involve actual contact. We can conceive of telekinesis, where someone moves something far away with their mind, or teleportation, where something disappears in one place and immediately appears somewhere else. But this is generally thought of either as supernatural magic, or else (as in Star Trek, I believe), done with very fast movement of litle particles and waves that isn’t quite instantaneous, and therefore has to travel, rather than actually acting at a distance.
When you hit a switch and turn on a light that’s far away, you actually connect an electric wire that’s attached to the switch, and electrons start to flow continuously along the wire, which includes the lightbulb, so there’s actually a physical chain of causation connecting the switch and the bulb, even though it goes a long distance very quickly. When you talk on a cell phone, your mouth causes the air to vibrate, which hits the wires in the cell phone in a way causing them to make electromagnetic rays, which travel in all directions until they hit a cell phone tower, which converts them into electric vibrations (or light waves, in fiber optic cables) that travel through wires to the phone on the other end of the line, where they turn back into air vibrations and hit the ear of the person listening. This seems like instantaneous action at a distance, but again, it’s all just very fast action that spreads quickly but continuously. No magic is needed.
In pre-modern physics, there was no action at a distance (I think) – Aristotle thought things moved up or down based on their intrinsic properties of what elements they were made of, and planets and stars were pushed by celestial beings, or just moved on their own accord.
Newton changed all this by postulating a gravitational force that things impose on one another instantaneously and at a distance. This was considered strange and perhaps even spooky, but it explained everything really nicely, so people got used to it. I believe the electric and magnetic fields also were introduced in this way, meaning that 19th century physics involved a whole lot of action at a distance. Two electrons exert a force on one another inversely proportional to the square of their distance, no matter how far away they are from one another, and that force changes instantaneously as they move closer or farther.
Einstein proposed his theory of relativity to reconcile some seeming imperfections in the equations describing electromagnetic fields of moving bodies. But once he made this change (actually, it might have needed to wait for general relativity), it also became clear that changes in electromagnetic and gravitational fields no longer propagated instantaneously, so that action at a distance was removed, restoring the intuitive order of things.
Quantum mechanics has threatened to bring back at least some forms of action at a distance though. Apparently, you can get two photons polarized in such a way that they have to be perpendicular. However, it’s not clear just what the two directions are until they’re measured. If they are simultaneously measured in the up-down direction, then one will pass and the other won’t. Same if they are simultaneously measured at 45 degrees to this direction. However, if one is measured up-down and the other is measured at 45 degrees, then there is no relation between the results of the measurement. But the point is, these particles may get very far from one another before they are measured, and they don’t know ahead of time which direction they’re going to be measured. If they’re both measured in the same direction (no matter which direction that is), they have to give opposite results, so it seems they have to instantaneously “communicate” with one another at a distance to make sure they make different “choices”. I believe this is where Bell’s theorem comes in, implying that there has to be some sort of “non-locality” in quantum mechanics, but I just don’t know quite what it actually says.
Anyway, I hope this is helpful.
You are profoundly out of touch with the revolution in the historiography of set theory. This work is changing our view of relativity, and it should certainly change yours.
Ryskamp, John Henry, “Paradox, Natural Mathematics and Twentieth-Century Ideas” (May 7, 2007). Available at SSRN: http://ssrn.com/abstract=897085