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  • "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




supervenes:

A mereology joke from my forthcoming dissertation.

supervenes:

A mereology joke from my forthcoming dissertation.


hi-res




I might be exaggerating a little if I say things like

  • We’re taught to measure our personal worth against exam scores;
  • We’re taught that there is One Competition and those who win the tournament get the goodies;
  • We’re taught that the children of Tiger Moms go to Yale and then Harvard Law and then become McKinsey consultants and then go on to head large corporations or i-banking or essentially win at life and rule the world in myriad ways;
  • We’re taught that the rest of us suck.

But I wouldn’t be completely making sh_t up. Those messages, or something like them, ∃ in the culture I come from and maybe in the culture you come from as well. Peter Thiel described a tournament to get into an Ivy League school, followed by a harder tournament to get into Stanford Law, followed by a harder tournament on Wall Street, … and left out of his story the 99.99% of us who didn’t even make it to the first tournament.

What about the supermajority? I’m pretty sure a hundred weak people can lift more weight than the strongest man on Earth. And I’m even more sure that the 50 smartest people on the planet can’t run Wall Street by themselves—let alone all the shops, shipyards, data centres, and engineering the runways of the airstrips to a millimetre of precision, that make up the economy.

 

So what about the rest of us? How much sense does it make to see the world in Thiel’s terms—the best versus the rest?

Well basic economics 101 tells us that a modern economy is made up of many specialised actors. The people who bend the tubes to make neon lights don’t know much about sewing shoes or sourcing the material for shoes, and none of those people know—or should know—how to do Ruby on Rails or Haskell.

Some people who research expertise also have developed a theory of 10,000 hours. If you practise something for 10^4 hours—so five years of work experience or ten years as a very, very consistent hobby—then you become awesome at it. A related theory is that if I have been doing something for a year or two and Peter Thiel tries to compete with me on it, I will still win regardless that he’s a chess master and a Stanford Law graduate and handsome and so on.

In other words, ∃ an equally or more compelling narrative than the A Player narrative: about everybody being different and that being okay and in fact more efficient.

Viewing education as a signalling mechanism to rank a one-dimensional hierarchy of best to worst people is one possibility—and one that BCG possibly uses to its advantage in applying profitable friction to the large companies who for some reason decide that some A+++ 24-year-olds know how to run their business better than they do. (Ooh, I really wanted to work in ‘fiction’ and ‘friction’ somehow. Too bad I was never a good enough student or I could have worked it.) But the dominant messages I hear from people who went into highly-paid frictional professions—accounting, law, consulting, finance—are that they want their kids to “find their own path”—i.e., do something with a tangible contribution to the society. Not necessarily fundraising for Laotian villagers, but something profitable that measurably increases the wealth of their community.

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So the “everyone is a special individual” message doesn’t just come from warmhearted Kindergarten teachers wearing seashell necklaces. If specialisation, difference, and diversity are more important than uniformly learning

  • the same parts of history,
  • the same mathematics,
  • and being compared to each other on a fabricated 7-dimensional scale (grades)
  • to see if we can get to be included in the golden inner circle of whatever mysterious ritual the white-shoe white-collar firms perform to add an order of magnitude more value to their customers per employee,

— then the hard-nosed economists are also telling us the same message. Maybe it is not about me being better than you and worse than Peter Thiel, but rather a high-dimensional poset network of symplectic skills and attributes, mostly not substitutable by smart people over dumb people and yet all worth pursuing as they complementarily add size to the world GDP.