Wild speculation: Perhaps one could also attempt to explain the effectiveness of mathematics the other way around: Why are the rules of physics simple enough so that we can describe them fairly accurately? I think this stems from the fact that complex systems are often the substrate for simpler systems. There are plenty of examples for this: Simple lifeforms emerge from biochemistry, planets are on elliptical orbits around stars as the result of countless particle interactions and so on. Perhaps this can be explained by the minimum total potential energy principle, that simple rules happen to be the ones which are stable or at least metastable and more complex systems would need more energy to maintain the relationships between its constituent elements. Simple systems can be abstracted from the substrate to the degree they are a reliable phenomenon. The human brain is a rather reliable information processing system, but it's also rather limited in capacity (it certainly has many times less capacity compared to the systems we are able to reason about). However, since the rules of most physical systems happen to be rather simple and since the language of mathematics allows us to compress rules into efficient chunks which fit into the working memories of the brightest humans, we are able to describe physical systems with counter-intuitive effectiveness.
Simple doesn't necessarily mean easy to understand, particularly when the thing doing the understanding is part of (and constrained by the rules of) the system being analysed. It is conceivable that there are simple aspects about our universe we will never understand because they are hidden behind some threshold of observation.
Exactly. I was trying to say that equivalently to "the unreasonable effectiveness of mathematics", we could also wonder at "the unreasonable simplicity of physical laws" (simple in the sense that we are able to understand the properties of it—at least superficially, but often also deeply). There is perhaps also an anthropic principle here: If the systems we find ourselves in would be much more complex, life would have likely been unable to form as prediction and organization would have been too difficult.
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.
When I mentioned I wouldn't be continuing with Economics after I graduated, one professor asked my why. I said that I had wanted to find a theory with the same mathematical elegance as classical economics (i.e. general equilibrium theory) but found that there didn't seem to be such a theory left to discover. Almost everything beyond classical economics was mathematically shallower (not to say it was wrong or poorly thought out, just that deeper mathematics wasn't applicable).
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 am highly skeptical whenever I see people trying to apply deep mathematics outside physics and chemistry
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/
It's okay to recommend entertaining content, but my impression was that Numb3rs in particular is quite shallow in how it applies math to solving crimes. Either I watched the wrong episodes, or they really just calculated what would be reasonably simple to deduce without the formulas.
is there something particular about Number3s that applies to this discussion? In general I like it when people explain to me what I don't know. Unfortunately I don't have time to watch the whole show.
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.
I can't continue a discussion with someone with your attitude. You're not reading what I'm writing carefully, or providing meaningful answers yourself. You've just classified me as someone who needs to be talked down to, and proceeded to talk down to me.
I also have a bad attitude to what you are saying - worse, in fact.
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.
Based on your replies to others presenting decent uses, I suspect you're not really interested in places where deep mathematics is used outside physics and chemistry, but there are many.
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 :)
You claim economics does not use deep math, so I will list some examples otherwise. Then perhaps you can tell me some areas you consider deep math that you think are not used in economics and I'll try to find places they are.
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?
It's probably not replying, because you are giving "evidence" which can easily verified, while I am making assertions based on my knowledge of the field, knowledge that you probably won't acknowledge I have because you're already convinced that I'm misguided on this topic. Anyway, all your examples (except for econometrics) are on the fringe of 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.
My take on this is, economy depends on human decisions whereas "pure" physics does not. To predict stock values even for one major company, with the greatest possible certainty you'd have to take into account everything that goes through the minds of the entire world population. And even that wouldn't be 100% precise because there would be unexpected events, such as natural disasters, probably even harder to predict than human decisions.
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!
The efficient market hypothesis states the value of a stock is the expected future dividends given all public information (or all private information, if you prefer the strong form).
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.
> the efficient market hypothesis assumes that emotion, mass psychology, etc. don't enter into stock markets
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.
no, according to the theory, "psychology" plays no role. People are assumed to have some information (signals) that inform them about the stock price. The efficient market hypothesis assumes that the stock price equals the expected value of discounted future dividends, conditional on all the signals of all the market participants (or conditional on all public signals, in the weak version).
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)
So, more generalized: economics is more focused on how the system moves towards equilibrium, but mostly ignores how the point of equilibrium itself fluctuates in time?
If anything it focuses more on the latter. Moving towards equilibrium isn't actually a meaningful concept in most of economics, since equilibrium is just the technical term for what happens when everyone acts in their own interest (e.g. Nash equilibrium) and correctly anticipates other actions (which in almost all economics models, does not require learning).
We just don't have a good enough model. Once (if) a good mathematical model that takes into account enough information is created, then the 'deep' mathematics that you speak of would once again be successful in predicting things.
We don't have a good enough model because the goals of economic theory are political, not explanatory.
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.
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.
This is partly what I'm arguing against (along with the premature belief that we already have a good enough model). I don't see any reason why there would ever be a model that didn't look like our existing models (say in economics) but with more bells and whistles. I don't think there will every be the equivalent of quantum field theory in economics.
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.
I would disagree - the economy is anti-inductive[1] (in that it resists attempts to understand it). As soon as you implement the results of your sufficiently-sophisticated model, the economy will start to be influenced by that.
Any viable model of the economy would therefore have to recursively contain a model of the model of the economy and so on.
see my other reply in this thread for deep vs shallow math. Yes, in some cases deep math is applicable to CS, e.g. dependent type theory, or the theory or Restricted Boltzman Machines applied to Deep Belief Networks.
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.
What? You have to be kidding me. TCS involves some of the 'deepest' mathematics known to mankind. Try convinvcing an expert that complexity theory and cryptography and type theory and PL theory aren't thoroughly mathematical. In fact computatbility theory and type theory lie at the very foundations of a reformulation of the basis mathematics in Homotopy Type Theory.
In fact, unlike physics and chemistry, which simply utilize deep mathematics, CS is deep mathematics.
>What? You have to be kidding me. TCS involves some of the 'deepest' mathematics known to mankind. Try convinvcing an expert that complexity theory and cryptography and type theory and PL theory aren't thoroughly mathematical. In fact computatbility theory and type theory lie at the very foundations of a reformulation of the basis mathematics in Homotopy Type Theory.
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.
First we need to define what computing science is. Computer Science is not software engineering. I agree that there is not much mathematics in software engg, but there is a significant amount of pioneering mathematics in say signals or cryptography or complexity theory. Moreover, even in the systems side of CS, there is a lot of math involved in say distributed systems or networks.
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.
This piece always irritates me. I don't think it's that interesting of a question.
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.
