algebraic mind

[gusl]

There is a cognitive style I like to call algebraic-mindedness. jcreed calls it "analogical mindedness".

I had just explained Landsburg's account of price discrimination in books: hard-cover vs soft-cover. Even though they cost the same to produce, and *everyone* prefers hard-cover, the producer has an interest in making soft-cover books, so that they can charge more from those who are willing to more pay more, while not losing those customers who are not willing to pay more. (This only works because copyright laws gives the producer a monopoly. I predict that ending copyright laws would drastically reduce the number of soft-cover books produced.)

Labour unions are the "dual" of corporate collusion to keep the salaries low (supposing this exists).

So he asked the analogical question: "what is the dual of price discrimination?"

There is no economic reason for asking this question, other than perhaps as a way to find out how to fight back against price discrimination.

The trait of algebraic-mindedness tends to lead one away from the original goal of the investigation, and towards improved understanding of the situation (a tighter knowledge network). The algebraic thinker is interested in general results: when looking at a specific situation, he wants to identify the minimum set of reasons that lead to the observed result. The algebraic thinker is the programmer who, when given a debugging task, insists on refactoring the code first. It is the physicist who will try to resolve paradoxes (philosophical progress) before making "ordinary" progress.

While making one slow at individual tasks, a reasonable amount of algebraic-mindedness tends to pay off on the long term: the algebraic thinker uses his numerous analogical bridges to "see" things that ordinary people can't. Math makes your smarter because it produces chunks of purely logical knowledge, and pure logic is extremely reusable: theorems can be instantiated on anything exhibiting the same mathematical structure. The algebraic thinker is the person who tries to extract the mathematical content out of ordinary situations, thereby creating this general knowledge.

While discussing topic A, algebraic thinkers T are known for bringing up apparently unrelated topics (let's call it B) which they know more about, because (1) A and B are analogically-linked in T's mind (2) T's knowledge of B lets him draw conclusions about A for free (or, in some cases, not for free (if the analogy is incomplete and needs to be verified), but still cheaper).

Questions algebraic thinkers tend to ask:
* if A and B are similar, why does A exhibit property X while B doesn't?

hmm... I'm thinking of splitting algebraic thinking into:
* analogical thinking
* mathematical thinking

The difference between these two kinds of thinking is that analogical thinking is about mappings between specific instances, while mathematical thinking is about mapping between abstract and concrete instances.

The mathematical idea of a situation X corresponds to the equivalence class of all possible situations analogous to X. (This type of analogy is an equivalence relation)

Comments

[ikeepaleopard]

1. It seems to me that you are just describing the process of generalizing from examples. In my opinion that is what all intelligent people do, though perhaps some are better at it than others.

2. I prefer paperback books, at least for fiction. They are easier to carry around.

[gustavolacerda.livejournal.com]

Using theorems about analogous situations is not the same thing as "generalizing from examples". The latter examples are all in the same given situation. Generalization would be finding a unified explanation for all instances (examples).

What I'm thinking of is: you observe A, analyze it to generalize your knowledge (the output of this analysis is a "proof" for a theorem with free variables, which explains the observation A). When analyzing B, you can just steal the above proof by plugging things in. You may optionally check its correctness again.

[ikeepaleopard]

So, what you are saying is that the difference between generalization and what you are talking about is a larger gap in the domains?

[jcreed.livejournal.com]

Well there are things going on at various levels.

Gustavo mentioned price discrimination. I noted that this is a phenomenon in which some amount of monopoly power allows a producer of goods some sort of pwnage-leverage over consumers of those goods. Generalization from examples is the step where I realize this is kind of like straightforward monopolies, which allow the producer to simply charge more across the board because of lack of competition. Price discrimination is a subtler situation where the producer can jigger prices to their advantage, because all competitors are collusively also so jiggering.

Now! Analogical reasoning is where I remember "hey, there are also such things as consumer boycotts, which are like exercises of monopoly (or oligopoly) power on the other end" and try to complete the analogy producer monopoly:price discrimination::consumer monopoly:?

The algebraic mindset is here just the fact that the analogy I'm hunting for is like looking for a dual, not just any old garden-variety analogue, and duality is obviously an algebraic sort of idea.

[gustavolacerda.livejournal.com]

Definition:

function f is a duality iff
for all x, f(f(x)) = x

Given a duality f, A is the f-dual of B, iff
f(A) = B and f(B) = A

[neelk]

Let's start with the simplest case of a monopolist engaging in perfect price discrimination, and dualize that. If you have a monopoly seller with the ability to set prices independently for each customer, the monopolist's optimal response is to set the marginal price of the good so that it exactly matches the marginal value each customer ascribes to it, thus capturing the entire surplus.

The dual case would be a single buyer (a monopsony), in which the buyer can offer a different price to each producer. Then, the optimal offer would be to offer to buy at a price that exactly matches the marginal costs of production for each seller. In this case, the economic surplus is captured entirely by the customer.

In both cases, monopoly/monopsony power can be can be subverted if the buyers/sellers can trade -- if buyers can trade with each other, then the monopolist can't offer a different price to each buyer because that creates arbitrage opportunities. For example, if Neel values proofs of cut-elimination at $10 dollars each, and Jason at $20 each, then a monopoly supplier of cut-elimination proofs (call him Kevin) can't charge Jason $20, because Neel can buy two at $10 and resell one to Jason.

Likewise, if sellers can trade with each other, the monopsonist can't offer a different price schedule to each seller, because that again creates opportunities for arbitrage.

[trufflesniffer.livejournal.com]

I think you're wise to make the split between analogical and mathematical thinking: The former is an attempt to fit two observed phenomena into a common logical schema; the latter is an attempt to present the former in a formal language.