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Posts tagged with ideation

[H]ow much of [Popper or Kuhn’s] philosophy relies upon actual, as opposed to reconstructed, history? Popper’s or Kuhn’s accounts of science bear little resemblance to actual science (and indeed if taken seriously I think both accounts would stop science dead in its tracks).

I also think, but this is very much my own view, that Kuhn imposed his notion of incommensurable revolutions on the history, and did not derive it from it. I think he came to the history with Wittgenstein and various other philosophical notions, and found what he was looking for. His historical work is excellent, but I do not think he derived his theoretical philosophy of science from it.

For a start, none of the supposed revolutions look anything like what he claimed they would. Even the Copernican “Revolution” takes over 200 years by his own admission.

It is my experience that when scientists appeal to Popper in particular, they manage to overlook that what they are trying to do scientifically is either ignored by Popper (like classification) or treated as irrelevant (like testability and verification, or discovery heuristics).

John S Wilkinson

(Source: evolvingthoughts.net)




[Alexander] Grothendieck expanded our … conception of geometry … by noticing that a geometric object 𝑿 can be … understood in terms of … maps from other objects into 𝑿.
Kevin Lin (@sqrtnegative1)

(Source: quora.com)




One of my projects in life is to (i) become “fluent in mathematics" in the sense that my intuition should incorporate the objects and relationships of 20th-century mathematical discoveries, and (ii) share that feeling with people who are interested in doing the same in a shorter timeframe.

Inspired by the theory of Plato’s Republic that “philosopher kings” should learn Geometry—pure logic or the way any universe must necessarily work—and my belief that the shapes
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a covering, drawn by Robert Ghrist www.math.upenn.edu/~rghrist
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and feelings thereof operate on a pre-linguistic, pre-rational “gut feeling” level, this may be a worthwhile pursuit. The commercial application would come in the sense that, once you’re in a situation where you have to make big decisions, the only tools you have, in some sense, are who you have become. (Who knows if that would work—but hey, it might! At least one historical wise guy believed the decision-makers should prepare their minds with the shapes of ultimate logic in the universe—and the topologists have told us by now of many more shapes and relations.)

To that end I owe the interested a few more blogposts on:

  • automorphisms / homomorphisms
  • the logic of shape, the shape of logic
  • breadth of functions
  • "to equivalence-class"

which I think relate mathematical discoveries to unfamiliar ways of thinking.

 

Today I’ll talk about the breadth of functions.

If you remember Descartes’ concept of a function, it is merely a one-to-at-least-one association. “Associate” is about as blah and general and nothing a verb as I could come up with. How could it say anything worthwhile?

The breadth of functions-as-verbs, I think, comes from which codomains you choose to associate to which domains.

The biggest contrast I can come up with is between

  1. a function that associates a non-scalar domain to a ≥0 scalar domain, and
  2. a domain to itself.

If I impose further conditions on the second kind of function, it becomes an automorphismThe conditions being surjectivity  and injectivity : coveringness ≥ 

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and one-to-one-ness 
≤  ↑
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successor function and square function
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Monotone and antitone functions  (not over ℝ just the domain you see = 0<x<1⊂ℝ)  These are examples of invertible functions.
.

If I impose those two conditions then I’m talking about an isomorphism (bijection) from a space to itself, which I could also call “turning the abstract space over and around and inside out in my hands” — playing with the space. If I biject the space to another version of itself, I’m looking at the same thing in a different way.

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Back to the first case, where I associate a ≥0 scalar (i.e., a “regular number” like 12.8) to an object of a complicated space, like

  • the space of possible neuron weightings;
  • the space of 2-person dynamical systems (like the “love equations”);
  • a space containing weird objects that twist in a way that’s easier to describe than to draw;
  • a space of possible things that could happen;
  • the space of paths through London that spend 90% of their time along the Thames;
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  • the space of possible protein configurations;
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then I could call that “assigning a size to the object”. Again I should add some more constraints to the mapping in order to really call it a “size assignment”. For example continuity, if reasonable—I would like similar things to have a similar size. Or the standard definition of a metric: dist(a,b)=dist(b,a); dist(x,x)=0; no other zeroes besides dist(self,self), and triangle law.

Since the word “size" itself could have many meanings as well, such as:

  • volume
  • angle measure
  • likelihood
  • length/height
  • correlation
  • mass
  • how long an algorithm takes to run
  • how different from the typical an observation is
  • how skewed a statistical distribution is
  • (the inverse of) how far I go until my sampling method encounters the farthest-away next observation
  • surface area
    File:Bronchial anatomy.jpg
  • density
  • number of tines (or “points” if you’re measuring a buck’s antlers)
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  • how big of a suitcase you need to fit the thing in (L-∞ norm)
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which would order objects differently (e.g., lungs have more surface area in less volume; fractals have more points but needn’t be large to have many points; a delicate sculpture could have small mass, small surface area, large height, and be hard to fit into a box; and osmium would look small but be very heavy—heavier than gold).

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Let’s stay with the weighted-neurons example, because it’s evocative and because posets and graphs model a variety of things.

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An isomorphism from graphs to graphs might be just to interchange certain wires for dots. So roads become cities and cities become roads. Weird, right? But mathematically these can be dual. I might also take an observation from depth-first versus breadth-first search from computer science (algorithm execution as trees) and apply it to a network-as-brain, if the tree-ness is sufficiently similar between the two and if trees are really a good metaphor after all for either algorithms or brains.

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imageBrains sound like a wicked-hard space to think about.  It’s a tightly connected (but not totally connected) network (graph theory)  Each of the nodes’ 3-D location may be important as well (voxels)  The signals propagate through time (dynamical)

More broadly, one hopes that theorems about automorphism groups on trees (like automorphism groups on T-shirts) could evoke interesting or useful thoughts about all the tree-like things and web-like things: be they social networks, roads, or brains.

 

So that’s one example of a pre-linguistic “shape” that’s evoked by 20th-century mathematics. Today I feel like I could do two: so how about To Equivalence-Class.

Probably due to the invention of set theory, mathematics offers a way of bunching all alike things together. This is something people have done since at least Aristotle; it’s basically like Aristotle’s categories.

  • The set of all librarians;
  • The set of all hats;
  • The set of all sciences;
  • Quine’s (extensional) definition of the number three as “the class of all sets with cardinality three”. (Don’t try the “intensional” definition or “What is it intrinsically that makes three, three? What does three really mean?” unless you’re trying to drive yourself insane to get out of the capital punishment.)
  • The set of all cars;
  • The set of all cats;
  • The set of all computers;
    Water Computer
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  • The set of all even numbers;
  • The set of all planes oriented any way in 𝔸³
  • The set of all equal-area blobs in any plane 𝔸² that’s parallel to the one you’re talking about (but could be shifted anywhere within 𝔸³)
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  • The set of all successful people;
  • The set of all companies that pay enough tax;
  • The set of all borrowers who will make at least three late payments during the life of their mortgage;
  • The set of all borrowers with between 1% and 5% chance of defaulting on their mortgage;
  • The set of all Extraverted Sensing Feeling Perceivers;
  • The set of all janitors within 5 years of retirement age, who have worked in the custodial services at some point during at least 15 of the last 25 years;
  • The set of all orchids;
  • The set of all ungulates;

The boundaries of some of these (Aristotelian, not Lawverean) categories may be fuzzy or vague

  • if you cut off a cat’s leg is it still a cat?
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    What if you shave it? What if you replace the heart with a fish heart?
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  • Is economics a science? Is cognitive science a science? Is mathematics a science? Is  Is the particular idea you’re trying to get a grant for scientific?

and in fact membership in any of these equivalence classes could be part of a rhetorical contest. If you already have positive associations with “science”, then if I frame what I do as scientific then you will perceive it as e.g. precise, valuable, truthful, honourable, accurate, important, serious, valid, worthwhile, and so on. Scientists put Man on the Moon. Scientists cured polio. Scientists discovered Germ Theory. (But did “computer scientists” or “statisticians” or “Bayesian quantum communication” or “full professors” or “mathematical élite” or “string theorists” do those things? Yet they are classed together under the STEM label. Related: engineers, artisans, scientists, and intelligentsia in Leonardo da Vinci’s time.)

But even though it is an old thought-form, mathematicians have done such interesting things with the equivalence-class concept that it’s maybe worth connecting the mathematical type with the everyday type and see where it leads you.

Characteristic property of the quotient topology

What mathematics adds to the equivalence-class concept is the idea of “quotienting” to make a new equivalence-class. For example if you take the set of integers you can quotient it in two to get either the odd numbers or the even numbers.

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  • If you take a manifold and quotient it you get an orbifold—an example of which would be Dmitri Tymoczko’s mathematical model of Bach/Mozart/Western theory of harmonious musical chords.
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  • If you take the real plane ℝ² and quotient it by ℤ²
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    (ℤ being the integers) you get the torus 𝕋²
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  • Likewise if you take ℝ and quotient it by the integers ℤ you get a circle.
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  • If you take connected orientable topological surfaces S with genus g and p punctures, and quotient by the group of orientation-preserving diffeomorphisms of it, you get Riemann’s moduli space of deformations of complex structures S. (I don’t understand that one but you can read about it in Introduction to Teichmüller theory, old and new by Athanase Papadopoulos. It’s meant to just suggest that there are many interesting things in moduli space, surgery theory, and other late-20th-century mathematics that use quotients.)
  • If you quotient the disk D² by its boundary ∂D² you get the globe S².
  • Klein bottles are quotients of the unit rectangle I²=[0,1]².

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So equivalence-classing is something we engage in plenty in life and business. Whether it is

  • grouping individuals together for stereotypes (maybe based on the way they dress or talk or spell),
  • or arguing about what constitutes “science” and therefore should get the funding,
  • or about which borrowers should be classed together to create a MBS with a certain default probabilities and covariance (correlation) with other things like the S&P.

Even any time one refers to a group of distinct people under one word—like “Southerners” or “NGO’s” or “small business owners”—that’s effectively creating an (Aristotelian) category and presuming certain properties hold—or hold approximately—for each member of the set.

File:Gastner map redblue byarea bystate.png
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File:Gastner map purple byarea bycounty.png
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File:Red and Blue States Map (Average Margins of Presidential Victory).svg

Of course there are valid and invalid ways of doing this—but before I started using the verb “to equivalence-class” to myself, I didn’t have as good of a rhetoric for interrogating the people who want to generalise. Linking together the process of abstraction-from-experience—going from many particular observations of being cheated to a model of “untrustworthy person”—with the mathematical operations of

  • slicing off outliers,
  • quotienting along properties,
  • foliating,
  • considering subsets that are tamer than the vast freeness of generally-the-way-anything-can-be

—formed a new vocabulary that’s helpfully guided my thinking on that subject.

Ordine geometrico demonstrata!




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 …

Jared Diamond, The World Before Yesterday

via University of David




One

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.
a larger &lsquot;thing&rsquot; with holes in it ... like the snow has &lsquot;road holes&rsquot; in it
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?

http://2.bp.blogspot.com/_sf5B7n5Avcg/THFKAnY0TCI/AAAAAAAAAmI/x9E4slMs0uQ/s1600/IMGP1553.JPG

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.




The origins of mass & the feebleness of gravity by Frank Wilczek

tl,dr:

  • dark matter & dark energy
  • "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.
  • Min 13:30 “the body and soul of QCD
    http://www.lightnessofbeingbook.com/LOBColorPlates/images/Plate01_ThreeJet.jpg
    http://www.fnal.gov/pub/today/images11/CMSResult042211figure1.jpg
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    img_lrg/jet.jpg not found

  • 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:
    Full-size image (27 K)

    http://www.yorku.ca/lewisr/images/Yspectrum.png
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    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.
    http://upload.wikimedia.org/wikipedia/commons/thumb/5/55/Bohr-atom-PAR.svg/500px-Bohr-atom-PAR.svg.png
    http://upload.wikimedia.org/wikipedia/commons/a/a4/HydrogenOrbitalsN6L0M0.png
  • "the highest form of musicality"
  • Quantum mechanics uses the mathematics of musical notes (vibrating harmonics).
  • Quantum chromodynamics uses the mathematics of chords, specifically triads since 3 colour forces act on each other at once.
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  • Particles are nothing more than stable tradeoffs that can be made between localisation costs (per energy) from QM and colour forces.
  • (Aside to quote Wikipedia: “Mathematically, QCD is a non-Abeliangauge theory based on a local (gauge) symmetry group called SU(3).”)

  • 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)
    http://www.grin.com/object/external_document.243440/7414ac37090135af37b2813c60318981_LARGE.png
    http://www-cdf.fnal.gov/physics/new/qcd/diphoton_2012/diphoton_xsec_10fb/plots/TXSec_massQLSSV01_07Jun12.gif
    http://cms.web.cern.ch/sites/cms.web.cern.ch/files/styles/large/public/field/image/Fig3-MassFactSoBWeightedMass.png
  • 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
  • Minute 67. Trips my regression-analysis circuits. Slopes & intercepts. Affine!
  • 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."

(Source: mit.tv)




  • "X does something whilst preserving a certain structure"
  • "There exist deformations of Y that preserve certain properties"
  • "∃ function ƒ such that P, whilst respecting Q"

This common mathematical turn of phrase sounds vague, even when the speaker has something quite clear in mind.

  

Smeet Bhatt brought up this unclarity in a recent question on Quora. Following is my answer:

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.

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In common mathematical abstractions, you might want to preserve a property like

after some transformation φ. 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.

 

A related idea is that of equivalence-class. If I make an equivalence class of all sets with cardinality 4, I’m talking about “their size is equivalent”.

Of course the set \texttt{ \{turkey, vulture, dove \} } is quite different to the set \{ \forall \texttt{ cones,\ the\ plane,\ a\ sheaf\ of\ rings} \} 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 ( \texttt{ things, distances\ between\ the\ things } ). (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:

  • Category: Structure-preserving mapsInvertible, structure-preserving maps

  • Groups: (group) homomorphism; (group) isomorphism
  • Rings: (ring) homomorphism; (ring) isomorphism
  • Vector Spaces: linear transformation, invertible linear transformation
  • Topological Spaces: continuous map; homeomorphism
  • Differentiable Manifolds: differentiable map; diffeomorphism
  • Riemannian Manifolds: conformal map; conformal isometry




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.
Maxwell's equations in differential form (reduces 20 to 4)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.
Gauss' law for rmagnetism
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)

Riemannian

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/>.

via University of David




I had nothing but ideas.

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.

Reuben Hersh, interviewed by John Brockman

emphasis mine. via davidaedwards

(Source: edge.org)