Math, mathematical models and metaphors

Source: Sam Harris podcast, date indeterminate

Interview with David Krakauer,  President – Santa Fe Institute

the m-cubed mayhem. That is m raised to the power of three mayhem. The mayhem comes from not understanding the difference between mathematics, the first m, mathematical models, the second m, and metaphors, the third.

what happens often is that arguments flare up because one person is using a term mathematically and another person metaphorically, and they don’t realize they’re doing this.

information is mathematically. It’s the reduction of uncertainty.

mathematically, the complex phenomena live somewhere between the regular and the random. Their hallmark signature is that their mathematical descriptions are long, and that’s what has made complex science so hard.

Complexity itself, as a concept, mathematically tries to capture how hard it is to describe a phenomenon. And as they get more complex, these descriptions get longer and longer.

That’s what we mean when we say someone is smart. They make things look easy.

complementary cognitive artifacts. Their unique characteristic is, not only do they augment your ability to reason in the form, for example, of multiplying and dividing, but when I take them away from you, you have in your mind a trace of their attributes that you can deploy.

when humans perform mathematics, they are using not the linguistic parts of their brain but the parts that represent number, which we share with nonhuman primates.

The abacus is a device for doing arithmetic in the world with our hands and eyes. But expert abacus users no longer have to use the physical abacus. They actually create a virtual abacus in the visual cortex. And that’s particularly interesting, because a novice abacus user like me or you thinks about them either verbally or in terms of our frontal cortex. But as you get better and better, the place in the brain where the abacus is represented shifts, from language-like areas to visual, spatial areas in the brain. It really is a beautiful example of an object in the world restructuring the brain to perform a task efficiently—in other words, by my definition, intelligently.

what I call competitive cognitive artifacts don’t so much amplify human representational ability as replace it. Another example that everyone is very enamored of now, rightly, is machine learning. We have this beautiful example recently of AlphaGo, a deep learning neural network being trained to beat an extraordinary ninth-dan Go player. That machine is basically opaque, even to its designers, and it replaces our ability to reason about the game. It doesn’t augment it.

if you become competent at the abacus, you’re not just competent at arithmetic. It actually has really interesting indirect effects on linguistic competence and geometric reasoning. It doesn’t have a firewall around it such that its functional advantages are confined to arithmetic. And in fact, I think that’s generally true for all interesting complementary cognitive artifacts.

A good example of this, which both Einstein and Frank Lloyd Wright depended upon, was wooden cubes. Early in their youth, they both became very enamored of these cubes and would construct worlds out of cubes, like Minecraft. And both of them claimed, Frank Lloyd Wright in the case of architecture and Einstein in the case of the geometry of the universe, that the intuitions they built up playing with these cubes were instrumental in their later lives. I would claim the same is true for maps. If you know how to navigate through a true space, like a Euclidean space or a curved space on the surface of the earth, that allows you to think about different kinds of spaces, relationship spaces, idea spaces. The notion of a path from one idea to another, as a metaphor, actually has an immediate and natural implementation in terms of a path in real space. You can see immediately how these things are of value more broadly.

What distinguishes a scientist from someone who has an orthodoxy? I guess it’s enshrined in Richard Feynman’s definition of a scientist as someone who believes in the ignorance of experts. That notion is the singular precondition for the possibility of science, which is a fundamental distrust in experts and expertise, including us. It has something to do with information, which as you pointed out, has something to do with uncertainty.

The optimistic future is the one where we say, “Enough. No more conformity, no more over-curation of what you think I should do and think.” A kind of radical accession of diversity, a radical individuality that we somehow reconcile with a constructive communitarian drive. I don’t think we’ve done that very well historically. How to be as different as we can be, but be congenial with one another. That is a positive future for me, but I think that’s the path of great labor.

I’m saying something slightly different, which is that the tools we now possess, which are so incredible, should be allowing us to have freedoms that are unprecedented …

Regardless, though, of whether or not there is life in the universe beyond our own planet, we have an intellectual obligation to populate it. That’s where I stand on the matter. Why do we do what we do? I think that if I have any kind of quasi-mythical belief system, it’s something to do with expanding the sphere of reason and sympathy into the world and beyond. If we could take the very best of what we’ve done and push it out into the universe, that would be an extraordinary thing.

If we could somehow make a commitment to the future more reflexive and more vivid, more emotionally and ethically salient to us, and internalize that—I think that’s one thing we need. I don’t know what it would look like, but the conjunction of your abacus talk and your saying it would be ethically problematic not to push forward into space in future generations made me think of this.

Harris: Who is your vote for the smartest person in human history? If you could put one human brain into the room to talk to the aliens, who would you nominate?

Krakauer: Probably John von Neumann.

Harris: That’s actually quite an uncontroversial pick, given what I know about him. But do you want to say a bit about why?

Krakauer: Well, the thing about Von Neumann that’s so incredible is that he created mathematical fields, physicals fields, computational fields, and social-scientific fields. That breadth and depth is almost unique to him. I wish I could pick several, but if I have to pick one, I pick him.

Harris: Just the stories about him, about the effect he had on the people around him, which included, arguably, more-famous and influential scientists and mathematicians—he was surrounded, as you know, by the most productive scientists of his generation. There are so many stories about the awe in which they held his ability to grasp and creatively interact with what they were doing, in real time, in a way that was just mesmerizing.

Krakauer: What was so incredible is that not only is there game theory, which he co-invented, and areas of quantum mechanics…

Harris: And the nuclear chain reaction stuff and meteorology….


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