Which mathematical ideas most influenced the way you think?

Question

This is not a question about how you use a formula or mathematical method to solve quantitative problems - that is applied mathematics. Rather, I'd like to hear how deeper ideas gained through the study of a particular mathematical field changed the way you think about more abstract problems or issues in the world outside us.

In other words, what mathematical field or idea, helped shape the way you think about non-mathematical situations?

Two examples of my own:

• Bayesian inference - The idea that even strong evidence is not enough to change an opinion, but rather that our willingness to accept evidence depends strongly on our prior position as well as our conviction in that position. This idea comes in handy any time you want to debate a strongly contested issue.

• Taylor series - The idea that complex phenomena can be described locally by a sum of terms, each less significant than the previous one. This helps think about complex tasks in "first order" vs. "second order terms, and helps to concentrate's ones efforts on the important things first.

Answer 1

For me, it's a simple but important (statistical) one- the gambler's fallacy.

This is the mistaken belief that if something happens more frequently than normal during some period, then it will happen less frequently in the future, or that if something happens less frequently than normal during some period, then it will happen more frequently in the future (presumably as a means of balancing nature).

Indeed, it's counterintuitive to think that, if you flip a fair coin 100 times and it shows up as heads every single time, that the probability of heads on the next term is exactly the same as it was the past hundred times (0.5).

A lot of people fall into this trap, especially rookie gamblers (hence the name), who may think that, after a 'bad day' (money lost), they are 'due' for some 'luck' (money gained), erroneously thinking that the universe is 'balanced'.

Answer 2

For me, simply understanding that every idea in math needs a proof in order to be justified changed the way I think. That idea has lead me to be more questioning of math itself, but also of the world and the status quo in a more general sense. Math also made it so I really latched onto Marx's idea's when I first encountered them.

Answer 3

A prominent cryptographer once told me that any pattern that distinguishes a cryptosystem from a random number generator possesses some internal symmetry which can then be exploited to create an attack. I'm not sure how exactly to word it, but there's some kind of life lesson in there: something about being mindful and observant, and learning to recognize the context surrounding every situation.

Answer 4

To do what you want, more is needed. For example, if you want to solve polynomial equations over the integers, your solutions lie outside the integers, or even the rationals. If you want to take the limits of Cauchy sequences of rationals, your limit is a real in general, not a rational. And in set theory, this is a big part of what large cardinals are all about: to prove what you want, you need the existence of stronger and stronger large cardinals. The passing to a larger domain of discourse happens constantly in mathematics.

EDIT: I should have credited this idea to Dana Scott with respect to his perspective on what large cardinals do for set theory.