Analogies always tickle me. They’re like bubbles: shiny, a little wobbly, fun to make, and even more fun to pop.

And for my money, physicists offer the best analogies.

They’re the best because they make no sense. They’re the best because they prompt you to go “Huh? What could that possibly mean?” They’re the best because, they show that math really is the language of the universe, and hilarious things happen when you try to translate the universe into English.

When I say “the universe,” I don’t mean black holes, quasars, and the whole cosmological freak show at the far edges of space. I’m talking about stuff we know. Matter. Light. Gravity. Electricity. Even these things, which we fold with a bored yawn into our most mindless routines… even these things are so damn weird that we can’t really describe them in plain English. We’ve got only two options: fancy math or gibberish analogies.

And given a choice between fancy math and gibberish analogies, you really can’t go wrong.

Electrons aren’t “its”. They have no individual identity. Feynman even had a somewhat half-baked theory that there is only one electron: it exists from the beginning of time to the end of time, then reverses course and exists from the end to the beginning, during which trip it looks like a positron, then forward in time again as an electron, and so on about 10^80 times.

I love that “one electron” theory. I’m sure there’s some fun, gibberish sci-fi out there arguing that the same is true of human souls. (By “gibberish sci-fi” I may mean “cult.”)

Is it really true that electrons have no individual identity? My understanding was that no two electrons can occupy precisely the same set of quantum states, and that they are quantized (you can’t have “half an electron”), which would suggest that they are in some sense discrete entities.

That’s true as far as it goes. But for example, suppose you have two atoms each of which emits an electron and absorbs an electron. The Fermi-Dirac statistics that describe the behavior of electrons (which is why they are called fermions) tells us that it makes no sense to ask “Did each atom absorb the same electron it emitted?” The case of atom A emitting an electron that is later absorbed by A and atom B emitting an electron that is later absorbed by B is simply not distinct from the case of A’s electron being absorbed by B and B’s electron being absorbed by A.

That is completely different from the behavior of pool balls, for example, which are individually numbered; they obey Maxwell-Boltzmann statistics (or would if you had enough of them), and could therefore be called “maxwellons” (though nobody says that). Normally, M-B statistics are used as a simplifying approximation to both F-D and Bose-Einstein (as in bosons) statistics in the limit of high temperature, where the particles are far enough apart to be effectively distinguishable.

“Imagine a great big soap bubble with one of those tiny little bubbles on the outside.”
“Okay.”
“Well it’s nothing like that.”
“Wait, so we’re in a tiny bubble universe sticking to the side of the bigger bubble universe?”
“Yeah. No! But if it helps, yes.”

Actually, we only have fancy math or gibberish analogies for mathematics, too.
What is this number line thingy, or the cartesian plane? Weird analogies. Only partly kinda related to numbers. Groups of objects to count? Once again, weird analogies, only partly kinda maybe related to the integers.
Hmmm. I think that I am going to teach counting to this group of five year olds. Should I start with the abstract algebra axioms? No, no, let me use set theory! That is the very easiest! Then they will learn to count correctly.
The number zero is number representing any set of which the members can be put into one-to-one correspondence with the members of the empty set.
The number one is the number representing any set of which the members can be put into one-to-one correspondence with the members of the set of the empty set.
The number two is the number representing any set of which the members can be put into one-to-one correspondence with the members of the set consisting of the empty set and the set of the empty set.
The number three is the number representing any set of which the members can be put into one-to-one correspondence with the members of the set consisting of the empty set, the set of the empty set, and the set of the empty set and the set of the empty set.
Do you understand? Ok, learn this over the weekend. On Monday, it is the real numbers. Dedikind cuts!

Well, that’s one way to do it. But Von Neumann’s natural numbers are easier than Frege’s. 0 is the empty set, 1 is the set {0} = {{}}, 2 is the set {0, 1} = {{}, {{}}}, 3 is the set {0, 1, 2} = {{}, {{}}, {{}, {{}}}, ….

Mathematics is a building in which you can replace the foundation at any time, leaving the superstructure unchanged.

I never encountered the way that Von Neuman defines the natural numbers, but I think that I like it.
That is why foundations is the best part of mathematics!
My favorite bedtime reading is Cantor, Dedkind, Hilbert, or Landau.

Good reading suggestions. Did Dedekind cuts way back when, but I haven’t explored other foundational stuff that much.

Also, John, “Mathematics is a building in which you can replace the foundation at any time, leaving the superstructure unchanged” is a very nice pull-quote.

That electron. What IS it even doing? Love the final illustration.

Thanks! That electron is totally a stand-in for me.

Most days I feel like the trail of footprints that Billy from Family Circus leaves behind: all loopy and confused.

Electrons aren’t “its”. They have no individual identity. Feynman even had a somewhat half-baked theory that there is only one electron: it exists from the beginning of time to the end of time, then reverses course and exists from the end to the beginning, during which trip it looks like a positron, then forward in time again as an electron, and so on about 10^80 times.

….Excuse me, I need to go clean my grey matter off of the floor over there.

I love that “one electron” theory. I’m sure there’s some fun, gibberish sci-fi out there arguing that the same is true of human souls. (By “gibberish sci-fi” I may mean “cult.”)

Is it really true that electrons have no individual identity? My understanding was that no two electrons can occupy precisely the same set of quantum states, and that they are quantized (you can’t have “half an electron”), which would suggest that they are in some sense discrete entities.

That’s true as far as it goes. But for example, suppose you have two atoms each of which emits an electron and absorbs an electron. The Fermi-Dirac statistics that describe the behavior of electrons (which is why they are called fermions) tells us that it makes no sense to ask “Did each atom absorb the same electron it emitted?” The case of atom A emitting an electron that is later absorbed by A and atom B emitting an electron that is later absorbed by B is simply not distinct from the case of A’s electron being absorbed by B and B’s electron being absorbed by A.

That is completely different from the behavior of pool balls, for example, which are individually numbered; they obey Maxwell-Boltzmann statistics (or would if you had enough of them), and could therefore be called “maxwellons” (though nobody says that). Normally, M-B statistics are used as a simplifying approximation to both F-D and Bose-Einstein (as in bosons) statistics in the limit of high temperature, where the particles are far enough apart to be effectively distinguishable.

That’s very cool. Adding “particle physics” to the list of things I need to read much, much more about.

Also, let’s work on bringing “maxwellons” into the popular lexicon.

I see. I think. Wait. No. No, I don’t see. Can you repeat that in just abstract math terms please? It makes more sense that way.

To paraphrase Dr. Who:

“Imagine a great big soap bubble with one of those tiny little bubbles on the outside.”

“Okay.”

“Well it’s nothing like that.”

“Wait, so we’re in a tiny bubble universe sticking to the side of the bigger bubble universe?”

“Yeah. No! But if it helps, yes.”

You two dudes need to team up:

Did BICEP2 Detect Gravitational Waves Directly or Indirectly?

http://profmattstrassler.com/2014/04/30/did-bicep2-detect-gravitational-waves-directly-or-indirectly/

Ah, my favorite kind of partnership! My partner brings the expertise, and I bring the not-expertise.

Actually, we only have fancy math or gibberish analogies for mathematics, too.

What is this number line thingy, or the cartesian plane? Weird analogies. Only partly kinda related to numbers. Groups of objects to count? Once again, weird analogies, only partly kinda maybe related to the integers.

Hmmm. I think that I am going to teach counting to this group of five year olds. Should I start with the abstract algebra axioms? No, no, let me use set theory! That is the very easiest! Then they will learn to count correctly.

The number zero is number representing any set of which the members can be put into one-to-one correspondence with the members of the empty set.

The number one is the number representing any set of which the members can be put into one-to-one correspondence with the members of the set of the empty set.

The number two is the number representing any set of which the members can be put into one-to-one correspondence with the members of the set consisting of the empty set and the set of the empty set.

The number three is the number representing any set of which the members can be put into one-to-one correspondence with the members of the set consisting of the empty set, the set of the empty set, and the set of the empty set and the set of the empty set.

Do you understand? Ok, learn this over the weekend. On Monday, it is the real numbers. Dedikind cuts!

Well, that’s one way to do it. But Von Neumann’s natural numbers are easier than Frege’s. 0 is the empty set, 1 is the set {0} = {{}}, 2 is the set {0, 1} = {{}, {{}}}, 3 is the set {0, 1, 2} = {{}, {{}}, {{}, {{}}}, ….

Mathematics is a building in which you can replace the foundation at any time, leaving the superstructure unchanged.

I never encountered the way that Von Neuman defines the natural numbers, but I think that I like it.

That is why foundations is the best part of mathematics!

My favorite bedtime reading is Cantor, Dedkind, Hilbert, or Landau.

Good reading suggestions. Did Dedekind cuts way back when, but I haven’t explored other foundational stuff that much.

Also, John, “Mathematics is a building in which you can replace the foundation at any time, leaving the superstructure unchanged” is a very nice pull-quote.

Haven’t read this yet, but…

http://bookre.org/reader?file=1232248

Ah, nice! My 12th-grade math teacher taught out of Spivak’s calculus book, and it was a great class. I’ve still got it sitting on my shelf.