The Unreasonable Effectiveness of Mathematics (1980)
dartmouth.edu
dartmouth.edu
It's actually hard to quantify simple systems so perhaps I should have replaced the word "most" with "many" in my comment above. There are also typos e.g. s/its/their/, sorry for that.
Why is there gravity and how does gravity pull are hard to explain, but it is easy to describe the degree of the pulling.
So while mathematics is very effective in Physics, and I believe it will remain so, I am highly skeptical whenever I see people trying to apply deep mathematics outside physics and chemistry. Some things just don't seem amenable to mathematical laws.
I don't understand exactly what you mean by deep mathematics but you should watch Numb3rs. And before I get yelled at for recommending a TV series when it comes to mathematics, here http://numb3rs.wolfram.com/
To answer you question, an example of shallow math is using an ODE to model population biology. This is mainly because the math doesn't actually tell you something that you qualitatively didn't already know. An example of deep mathematics is using representation theory to analyze the quantum charge, spin etc of fundamental particles.
Yes, it enumerates real-world applications of mathematics in fields other than Physics/Chemistry, of which you said you were skeptical.
This is mainly because the math doesn't actually tell you something that you qualitatively didn't already know.
So I take it that the depth of mathematics is determined by its ability to predict?
You say physics and chemistry are the only deep uses of math, and illustrate that with a physics example to define "deep", and a biology example to define "shallow". That's not meaningful at all: it's circular logic (i.e. bad mathematics).
Without math, we would have a much poorer understanding of population biology. Your population biology example was from 1926 and the state-of-the-art has moved on - so you have badly represented the subject. I suggest that your personal understanding of biology is where the shallowness really lies.
The parent commenter has made a genuine effort to understand you and offer you some advice, despite your bullshit. You should thank them.
oooookay.
I'm basing this on having tried to find it myself, and becoming convinced that there is no such theory, and also that the rest of the field is correct in not trying to find such a theory.
Any viable model of the economy would therefore have to recursively contain a model of the model of the economy and so on.
> As soon as you implement the results of your sufficiently-sophisticated model, the economy will start to be influenced by that.
This is called a feedback loop, systems with feedback loops are well understood.
Economic 'explanations' exist to privilege certain actors through pseudo-rational justifications for personal benefit, not to predict economic reality.
If you look objectively at economic models, you'll see that all the mainstream models are uniformly hopeless at predicting the future, and some of the most highly placed analysts have absolutely no insight into future events - not just in defiance of theory, but often in defiance of simple common sense.
See e.g.
http://www.bbc.co.uk/news/business-30699476
This is characteristic of a pseudoscience, not a real science.
The only way to profit from this is to buy government policy and to fix the markets - something the bigger players are well aware of, and have been doing enthusiastically for decades now.
For some reason the theoretical economics is, instead, trying hard to ignore the human factor and squeeze patterns out of what's happening in the economy, i.e. the system is viewed pretty much like an isolated thermodynamic one. From this perspective a human being is merely an automaton that buys and sells stuff, just like molecules in thermodynamics are just little thingies with a few simple physical properties. Except humans are obviously far more complicated than that!
This is a good example of how academic economics think because on the one hand it is consistent with your claim that "you'd have to take into account everything that goes through the minds of the entire world population" and on the other hand it "ignores the human factor" in that the efficient market hypothesis assumes that emotion, mass psychology, etc. don't enter into stock markets.
I think economists actually ignore the "human factor" in the right way, in order to get some reasonable approximation of reality. The problem is that adding the human factor back in doesn't result in a mathematically elegant theory, just a more complicated model. So they are unable to progress in the same way as physics can.
Isn't mass psychology part of "public information"?? I know little about stock markets, but from the outside I would say that "how I think people will feel about a company" is pretty much what I would base my "bet" on whether a stock will go up or down.
It's quite different to how you describe you imagined the stock markets working. But I also believe it's a more accurate picture overall. Markets are pretty accurate at least with regard to individual stocks (see Robert Schiller's work for a non-mainstream but still reasonable theory of irrational exuberance, where all stocks in the market might be over/under-valued at a given time)
In many cases, my original point applies: in practice the math needed is shallow, and people only use deep math because they want to, or because they want to publish papers.
In fact, unlike physics and chemistry, which simply utilize deep mathematics, CS is deep mathematics.
Once upon a time I had a research mentor who kept the following limerick on his academic jokes page. I find it applicable here to explain why people think Theoretical CS isn't mathematical:
No idea is too obvious or dreary, If appropriately expressed in type theory. It's a research advance, no-one understands, But they are too impressed to be leery.
Or in other words, actual theoretical CS, which does employ deep mathematics, is a tiny field compared to most of what people consider "Computing Science" to be.
Computer Science is a field FOUNDED on the deepest parts of mathematics, so I find any claim that CS is not a fundamentally mathematical discipline a joke.
>Yes, in some cases deep math is applicable to CS, e.g. dependent type theory
Homotopy type theory is an extension/variation of dependent type theory.
I suspect you didn't study enough economics - there are certainly areas where the math gets deep (PhD in math here, algebraic geometry, so I know what deep math means).
Given this forum, the obvious one is CS, but you already tried to ignore that one, even though the math used includes all manner of graph theory (which gets very deep), number theory (especially crypto, which is a driving force in some areas of the pure math), and so on.
Another is finance (stochastic calculus to start, and a whole host of other disciplines). There is a lot of decently hard math down this route.
Another is medicine or any science where evidence is hard to tease out (statistics, which again gets quite deep, and professional research statisticians often are used as consultants since the uses are not simply plugging data into known formulas).
There are others, but it will be interesting to see how this short list goes :)
I studied more than enough, and more than you. Why don't you tell me the areas of economics you think involve deep mathematics?
Kantorovich and Koopmans (Nobel Econ) claimed Dantzig (a very pure mathematician, invented simplex algorithm) deserved to share their Nobel Prize for linear programming - they were using cutting edge mathematics, right?
Calculus of variations, optimal control, dynamic programming - used extensively and many areas extended for economic questions (Ramsey, Hotelling). Lots of new math invented for these uses.
Von Neumann introduced functional analytic methods and topology in order to prove his generalization of Brouwer's fixed-point theorem in order to prove existence of optimal equilibrium for his model of economic growth - that is a pure math result, and a nice one, developed for an economic problem. Any generalization of the current best generalization of Brouwer's fixed point theorem is a very nice result. VonNeumann did a lot of other math work for his economic interests.
Nash similarly.
Smale of Field medal fame used Baire category theory and Sard's lemma to establish a new method of obtaining general equilibria for economic questions. Do those count as deep?
Game theory has a host of methods from all over mathematics to prove new results, using new mathematical results derived solely to prove results in game theory. Grab a monograph and see if any of this is deep enough for you.
Econometrics uses all level of statistics to answer measurement questions. Again, the tools and new results are there.
I could go on, but now it's your turn. What math is deeper by your definition than all the pieces already used/needed/invented for current use in economics?
Linear programming is almost never used, since economists like exact analytic solutions. Maybe it was popular when that Nobel prize was given?
Calculus of variations, optimal control, dynamic programming: yes, although I would not call these applications of deep math. The underlying ideas are very simple, they've just been written in much more general forms for reasons that are unclear to me. But only the simple versions get used. Same applies to fixed point theorems. You might learn and forget these things in the first two years of a PhD program, but that's about it for 99% of economists.
>Smale of Field medal fame used Baire category theory and Sard's lemma to establish a new method of obtaining general equilibria for economic questions. Do those count as deep?
Yes, and almost certainly useless for econ. But maybe you can explain a bit about it.
>Game theory has a host of methods from all over mathematics to prove new results, using new mathematical results derived solely to prove results in game theory. Grab a monograph and see if any of this is deep enough for you.
I'm very familiar with game theory. But thanks for the advice. In return, I'd suggest you read the required reading for the first and second years of a PhD program. Something like graduate Micro and Macro. You might still consider the math deep, but you won't see any category theory or algebraic topology, and existence and uniqueness theorems will be relegated to appendices in fine print.
Firstly, Mathematics is a language for studying structure.
The reason it works is because we won't call something "mathematics" unless it is a reliable/reproducible/communicable tool for representing structure. As it is a language, it employs linguistic structure in an act of mimicry of experience. This is called analogy.
Secondly, the brain -- the basic function of a brain -- is to differentiate between experiences -- to assign meaning or to separate a signal from noise. This is how a brain develops a notion of 'appropriateness.' We apply this concept to linguistic analogies, and from that you can recover a notion of 'truth.' (An 'appropriateness engine' could be a suitable term for a moral algorithm. 'Utility function' is also often used in this context.)
It is not unreasonable that math works. Math works because if some language doesn't work, we just won't call it math. The mystery here is that anything works at all, and to answer that you'd have to explain why anything even exists.