But there were also more profound features, which took me a long time even to notice, because they are so at odds with modern experience that neither New Guineans nor I could even articulate them. Each of us took some aspects of our lifestyle for granted and couldn’t conceive of an alternative.
Those other New Guinea features included the non-existence of “friendship” (associating with someone just because you like them), a much greater awareness of rare hazards, war as an omnipresent reality, morality in a world without judicial recourse, and a vital role of very old people.
…
Many of my experiences in New Guinea have been intense—a sudden encounter at night with a wild man, the prolonged agony of a nearly-fatal boat accident, one broken little stick in the forest warning us that nomads might be about to catch us as trespassers …
Counting generates from the programmer’s successor function ++ and the number one. (You might argue that to get out to infinity requires also repetition. Well every category comes with composition by default, which includes composition of ƒ∘ƒ∘ƒ∘….)
But getting to one is nontrivial. Besides the mystical implications of 1, it’s not always easy to draw a boundary around “one thing”. Looking at snow (without the advantage of modern optical science) I couldn’t find “one snow”. Even where it is cut off by a plowed street it’s still from the same snowfall. And if you got around on skis a lot of your life you wouldn’t care about one snow-flake (a reductive way to define “one” snow), at least not for transport, because one flake amounts to zero ability to travel anywhere. Could we talk about one inch of snow? One hour of snow? One night of snow?
Speaking of the cold, how about temperature? It has no inherent units; all of our human scales pick endpoints and define a continuum in between. That’s the same as in measure theory which gave (along with martingales) at least an illusion of technical respectability to the science of chances. If you use Kolmogorov’s axioms then the difficult (impossible?) questions—what the “likelihood” of a one-shot event (like a US presidential election) actually means or how you could measure it—can be swept under the rug whilst one computes random walks on trees or Gaussian copulæ. Meanwhile the sum-total of everything that could possibly happen Ω is called 1.
With water or other liquids as well. Or gases. You can have one grain of powder or grain (granular solids can flow like a fluid) but you can’t have one gas or one water. (Well, again you can with modern science—but with even more moderner science you can’t, because you just find a QCD dynamical field balancing (see video) and anyway none of the “one” things are strictly local.)
And in my more favourite realm, the realm of ideas. I have a really hard time figuring out where I can break off one idea for a blogpost. These paragraphs were a stalactite growth off a blobular self-rant that keeps jackhammering away inside my head on the topic of mathematical modelling and equivalence classes. I’ve been trying to write something called “To equivalence class” and I’ve also been trying to write something called “Statistics for People Who Program Computers” and as I was talking this out to myself, another rant squeezed out between my fingers and I knew if I dropped the other two I could pull One off it could be sculpted into a readable microtract. Leaving “To Equivalence Class”, like so many of the harder-to-write things, in the refrigerator—to marinate or to mould, I don’t know which.
But notice that I couldn’t fully disconnect this one from other shared-or-not-shared referents. (Shared being English language and maybe a lot of unspoken assumptions we both hold. Unshared being my own personal jargon—some of which I’ve tried to share in this space—and rants that continually obsess me such as the fallaciousness of probabilistic statements and of certain economic debates.) This is why I like writing on the Web: I can plug in a picture from Wikipedia or point back to somewhere else I’ve talked on the other tangent so I don’t ride off on the connecting track and end up away from where I tried to head.
The difficulty of drawing a firm boundary of “one” to begin the process of counting may be an inverse of the “full” paradox or it may be that certain things (like liquid) don’t lend themselves to counting in an obvious way—in jargon, they don’t map nicely onto the natural numbers (the simplest kind of number). If that’s a motivation to move from discrete things to continuous when necessary, then I feel a similar motivation to move from Euclidean to Hausdorff, or from line to poset. Not that the simpler things don’t deserve as well a place at the table.
We thinkers are fairly free to look at things in different ways—to quotient and equivalence-class creatively or at varying scales. And that’s also a truth of mathematical modelling. Even if maths seems one-right-answer from the classroom, the same piece of reality can bear multiple models—some refining each other, some partially overlapping, some mutually disjoint.
“Even though protons, neutrons, and electrons comprise only 3% of the universe’s mass as a whole, I hope you’ll agree that it’s a particularly significant part of the mass.” lol
“Just because you can say words and they make sense grammatically doesn’t mean they make sense conceptually. What does it mean to talk about ‘the origin of mass’?”
“Origin of mass” is meaningless in Newtonian mechanics. It was a primitive, primary, irreducible concept.
Conservation is the zeroth law of classical mechanics.
F=MA relates the dynamical concept of force to a kinematic quantity and a conversion factor (mass).
rewriting equations and they “say” something different
the US Army field guide for radio engineers describes “Ohm’s three laws”: V=IR, I=V/R, and a third one which I’ll leave it as an exercise for you to deduce”
m=E/c²
Einstein’s original paper Does the inertia of a body depend on its energy content? uses this ^ form
You could go back and think through Einstein’s problem (knowing the solution) in terms of free variables. In order to unite systems of equations with uncommon terms, you need a conversion factor converting a ∈ Sys_1 to b ∈ Sys_2.
Protons and neutrons are built up from quarks that are moving around in circles, continuously being deflected by small amounts. (chaotic initial value problem)
supercomputer development spurred forward by desire to do QCD computations
Min 25:30 “The error bounds were quite optimistic, but the pattern was correct”
A model with two parameters that runs for years on a teraflop machine.
Min 27:20 The origin of mass is this (N≡nucleon in the diagram): QCD predicts that energetic-but-massless quarks & gluons should find stable equilibria around .9 GeV:
Or said alternately, the origin of mass is the balance of quark/gluon dynamics. (and we may have to revise a bit if whatever succeeds QCD makes a different suggestion…but it shouldn’t be too different)
OK, that was QCD Lite. But the assumptions / simplifications / idealisations make only 5% difference so we’ll still explain 90% of the reason where mass comes from.
Computer ∋ 10^27 neutrons & protons
The supercomputer can calculate masses, but not decays or scattering. Fragile.
Minute 36. quantum Yang-Mills theory, Fourier transform, and an analogy from { a stormcloud discharging electrical charge into its surroundings } to { a "single quark" alone in empty space would generate a shower of quark-antiquark virtual pairs in order to keep a balanced strong charge }
Minute 37. but just like in QM, it “costs” (∃ a symplectic, conserved quantity that must be traded off against its complement) to localise a particle (against Heisenberg uncertainty of momentum). And here’s where the Fourier transform comes in. FT embeds a frequency=time/space=locality tradeoff at a given energy (“GDP” in economic theory). The “probability waves” or whatever—spread-out waveparticlequarkthings—couldn’t be exactly on top of each other, they’ll settle in some middle range of the Fourier tradeoff.
“quasi-stable compromises”
This is similar to how the hydrogen atom gets stable in quantum mechanics. Coulomb field would like to pull the electron on top of the proton, but the quantum keeps them apart.
Minute 40. Because the compromises can’t be evened out exactly due to quanta, there’s some leftover energy. It’s the same for a particular kind of quark-gluon interaction (again, because of the quanta). The .9 GeV overshoot | disbalance | asymmetry in some particular quark-gluon attempts to balance creates the neutrons and protons. And that’s the origin of mass.
Minute 42. Feebleness of gravity.
(first of all, gravity is weak—notice that a paperclip sticks to a magnet rather than falling to the floor)
(muscular forces are the result of a lot of ATP conversions and such. That just happens to be even weaker—but if you think of how far removed those biochemical electropulses and cell fibres are from the fundamental foundation, maybe that’s not so surprising.)
Gravity is 40 orders of magnitude weaker than the electrical force. Not forty times, forty orders of magnitude.
Planck’s vision; necessary conversion; a theory of the universe with only numbers.
The Planck distance, even for nuclear physicists, is about 20 orders of magnitude too small.
The clunkiness of Planck’s constants mocks dimensional analysis. “If you measure natural objects in natural units, you should get something of the order of unity”.
“If you agree that the proton is a natural object and the Planck scale is a natural unit, you’d be off by 18 orders of magnitude”.
Suppose gravity is a primitive. Then the question becomes: “Why is the proton so light?” Which now we can answer. (see above)
Simple physics (local interactions, basic = atomic = fundamental = primitive behaviours) should occur at Planck scales. (More complex behaviours then should “emerge” out of this reduction.)
So that should be, in terms of energy & momentum, 10^18 proton masses, where the fundamental interactions happen.
The value of the quark-gluon interaction at the Planck scale. “Smart” dimensional analysis says the quantum level that makes protons from the gluon-quark interactions then gets us to ½, “which I hope you’ll agree is a lot closer to unity than 10^−18”.
Minute 57. “A lot of what we know about the deep structure of the Standard Model is summarised on this slide”
weak force causes beta decay
standard model not so great on neutrino masses
SO(10)’s spinor representation has all the standard model’s symmetries as subgroups
Supersymmetry would have changed the clouds and made everything line up real nicely. (The talk was in 2004 and this week, in 2012, the BBC reported that SuSy was kneecapped by the latest LHC evidence.)
“If low-energy supersymmetry turns out to be false, I’ll be very disappointed and we’ll have to think of something else.”
It depends on the category. The idea of isomorphism varies across categories. It’s like if I ask you if two things are “similar” or not.
“Similar how?” you ask.
Think about a children’s puzzle where they are shown wooden blocks in a variety of shapes & colours. All the blocks that have the same shape are shape-isomorphic. All the blocks of the same colour are colour-isomorphic. All the blocks are wooden so they’re material-isomorphic.
In common mathematical abstractions, you might want to preserve a property like
after some transformationφ. It’s the same idea: “The same in what way?”
As John Baez & James Dolan pointed out, when we say two things are “equal”, we usually don’t mean they are literally the same. x=x is the most useless expression in mathematics, whereas more interesting formulæ express an isomorphism:
“Something is the same about the LHS and RHS”.
“They are similar in the following sense”.
Just what the something is that’s the same, is the structure to be preserved.
Of course the set is quite different to the set in other respects. Again it’s about “What is the same?” and “What is different?” just like on Sesame Street.
Two further comments: “structure” in mathematics usually refers to a tuple or a category, both of which mean “a space” in the sense that not only is there a set with objects in it, but also the space or tuple or category has mappings relating the things together or conveying information about the things. For example a metric space is a tuple . (And: having a definition of distance implies that you also have a definition of the topology (neighbourhood relationships) and geometry (angular relationships) of the space.)
In the case of a metric space, a structure-preserving map between metric spaces would not make it impossible to speak of distance in the target space. The output should still fulfill the metric-space criteria: distance should still be a meaningful thing to talk about after the mapping is done.
I’ve got a couple drafts in my 1500-long queue of drafts expositing some more on this topic. If I’m not too lazy then at some point in the future I’ll share some drawings of structure-preserving maps (different “samenesses”) such as the ones Daniel McLaury mentioned, also on Quora:
It wasn’t Einstein, but the mathematician Hermann Weyl who first addressed the [distinction] [between gravitational and non-gravitational fields] in 1918 in the course of reconstructing Einstein’s theory on the preferred … basis of a “pure infinitesimal geometry”….
Holding that direct…comparisons of length or duration could be made at near-by points of spacetime, but not … “at a distance”, Weyl discovered additional terms in his expanded geometry that he … formally identified with the potentials of the electromagnetic field. From these, the electromagnetic field strengths can be immediately derived. Choosing an action integral to obtain both [sorts of] Maxwell equations as well as Einstein’s gravitational theory, Weyl could express electromagnetism as well as gravitation solely within the confines of a spacetime geometry. As no other interactions were definitely known to occur, Weyl proudly declared that the concepts of geometry and physics were the same.
Hence, everything in the physical world was a manifestation of spacetime geometry. (The) distinction between geometry and physics is an error, physics extends not at all beyond geometry: the world is a (3+1) dimensional metrical manifold, and all physical phenomena transpiring in it are only modes of expression of the metric field, …. (M)atter itself is dissolved in “metric” and is not something substantial that in addition exists “in” metric space. (1919, 115–16)
Ryckman, Thomas A., “Early Philosophical Interpretations of General Relativity”, The Stanford Encyclopedia of Philosophy (Fall 2012 Edition), Edward N. Zalta (ed.), forthcoming URL = <http://plato.stanford.edu/archives/fall2012/entries/genrel-early/>.
O.K., they weren’t strictly mine, in the sense that these ideas were acquired, arranged, styled, photographed, published and distributed by entities bearing no relation to me whatsoever.
What is mathematics? It’s neither physical nor mental, it’s social. It’s part of culture, it’s part of history.
It’s like law, like religion, like money, like all those other things which are very real, but only as part of collective human consciousness. That’s what maths is.
is quite different to the set 
in other respects. Again it’s about “What is the same?” and “What is different?” just like on 
. (And: 
Choosing an action 
