Several people have asked me whether studying math changed my view of the world, and if so how.

I see applications of math everywhere. But more fundamentally, studying math has led me to believe that complex problems sometimes have simple solutions.

Simple solutions may be hard or impossible to find. But you’re more likely to find a simple solution if you believe it exists because you’ll keep looking longer.

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As I grow older, math encourages me to find patterns underneath things. While I was very much an agnostic when younger, now I’m almost a Platonist. Maybe it’s my own head looking for patterns in noise and finding them …. But I am amazed at what appears to be the “unreasonable power” of mathematics to find answers and bring order to things which, on surface, appear not to say anything at all.

My personal biggest revelation was understanding was just a language. Like human languages it lets me express thoughts and ideas. If I can’t put something in to a mathematical form chances are I haven’t thought it out well enough. After that the semantics and syntax of mathematical transformations can help guide me toward a solution, often when the solution isn’t apparent from the English language description of the problem.

Math is a method of putting the abstractions and interpretations of our world into a concrete and readable form. It allows humans to both interpret and predict their surroundings. It has influenced my thoughts greatly in how I view reality as a model. Everyone models every aspect of their lives and these models, in theory, can all be laid down mathematically even if drastic approximations are needed. All of life is a model and there is no better way to represent and communicate that model than through mathematics.

The paradoxes in logic and set theory have really shaken my confidence that the world is basically intelligible to us. There is no consensus for how those paradoxes should be resolved, and the methods out there are never completely satisfying. (I think you can show that many of the proposed solutions, including ZF, themselves generate new philosophical problems.) If contradictions appear in the foundations of mathematics, then it seems likely that we cannot achieve a self-consistent picture of the world.

I’m not as concerned about the foundations of mathematics as the modeling assumptions that go into the application of mathematics. The weakest link may be the first one: “Assume …”

For example, people have gone to prison because someone wrongly assumed two probabilities were independent. That’s the kind of thing that worries me.

In the statistical realm, the weakest link may be the

unstatedassumptions. That was a big eye-opener for me. I’m not familiar with the court cases you mention, but I doubt the statistical analysis said anything about independence. They probably performed an analysis which was only valid if the two probabilities were independent, but didn’t even consider whether they were or not.I recall a seminar in which it was pointed out that in earlier days parternity cases were settled partly on data analysis which assumed that the paternity candidate had an a priori probability of being the biological father of 50%. This assumption wasn’t made explicitly, and on further reflection from a Bayesian point of view it certainly wouldn’t be warranted in all cases.

My favorite example comes from a newspaper article reporting that two different US states had idential winning lottery numbers on the same day. They were both of the ‘pick six numbers from 1 to 50″ variety. The article reported that the probability of this occuring was on the order of 1 in a quadrillion, which is incredible. Of course it was no less likely than someone winning one of these lotteries with a single ticket, which although extremely unlikely has been observed.

I sometimes wonder about p-values, and that most people observing a p-value of 10^-6 would take it as being basically iron-clad proof while simultaneously thinking that winning the lottery is likely enough to warrant buying a ticket. We are funny critters, we humans.

For me I think it has simply made me more patient and methodical in thinking through things than if I didn’t have any mathematical training. By studying math I’ve learned that problems which at first appear extremely complicated and forbidding can often be solved very well … you just need to find the right way to think about them.

But now that I think about it, I’m realizing I don’t apply this patient problem solving mentality as much as I could to goals in my personal life (e.g. losing weight,saving money etc.) – maybe I should more consciously try to do that…

As someone on the periphery (social science statistician), I’ve noticed anecdotally that mathematicians are unusually prompt about updating their beliefs in the face of contradictory evidence. YMMV.

Ever since logarithboo**ms, I was interested in the idea of a universal rythem. Since then I have discovered it : the bodidlidleyrithm (bo) which is n=(1/log(bo))**9

The rest history…

Sluggowski

John,

Your comment about faith in the existence of a simple answer reminds me of Paul Erdős‘s “Book”.

Just the faith that there is an answer reminds me of a group of folks whose avocation is deciphering ancient ciphertexts. According to a story I heard, Jim Reeds had in unshakeable faith that there was a method used in constructing the tables in the Book of Soyga which motivated him to discover it at long last. One of the guys trying to decipher the Voynich Manuscript states that this is great encouragement in his own work.