but which still require testing to verify

Today, we do experiments in our heads. The University of Houston's College of Engineering presents this series about the machines that make our civilization run, and the people whose ingenuity created them.

The late 19th-century German philosopher Ernst Mach added a phrase to our language. It was "thought experiment." It's a wonderfully contradictory idea.

One scientist observes and measures nature. Another writes mathematical theories. Experiment and thought seem to be separate roads to learning. Mach walked the experimenter's road. But he said we can also experiment in our heads, not just in nature.

He said, look what Galileo did. Aristotle thought heavy things should fall faster than light ones. He thought a stone should fall faster than a feather. So Galileo said, "Let's glue a feather to a stone and drop it. Together, they weigh more than either the feather or the stone alone. If Aristotle's right, they'll fall faster together than either would fall alone. But who could believe the feather wouldn't slow the stone? Aristotle has to be wrong."

That's what we call a thought experiment. Galileo did the experiment in his head. Mach made a funny argument to justify doing that. He said our intuition is accurate because evolution has shaped it. We shrink from the absurd. That's what Galileo was doing when he said a feather can't speed the fall of a stone.

Enter now the grand master of thought experiments, Albert Einstein. Einstein was Mach's disciple. He created a great theater of thought experiments to explain relativity. But relativity flies in the teeth of our intuition. He talked about throwing balls and reading clocks on railroad trains and elevators. His conclusions had nothing to do with intuition.

Mach finally cracked under the weight of Einstein's admiration. Relativity theory was, he wrote, "growing more and more dogmatic." For Mach, a theory was only a framework for our measurements. Einstein figured that you first wrote correct theories. Then the data would simply fall into place.

In 1919 Einstein shrugged off an important telegram. Someone wired that they'd seen rays of light bend under the influence of gravity. That's what relativity predicts. Einstein was unimpressed. If light hadn't bent, he said, then "I'd have been sorry for the dear Lord. The theory is correct!"

So Einstein took Mach's thought experiments and used them in ways Mach never intended. He used them to show us a world that's far more subtle and surprising than anything our intuition ever prepared us for. Mach's ideas about thought experiments finally faded because the world is much stranger than our capacity to imagine it.

so it turns out, there's actually a pretty easy proof that if f=f' and f is analytic, then f=ce^x for some constant c

i always suspected that f=f' meant either f=0 or f=ce^x, but i never saw such an elegant proof of uniqueness

also makes sense why it's basically the way to solve differential equations nontrivially, as having an eigenfunction seems pretty durn important



probably isn't special to people who are smart

i'm not certain though about the truth of this statement outside of analytic functions, nor about the analyticity of solutions to differential equations

On second thought, it's not particularly surprising after realizing how Taylor expansion works. Don't care, insights are insights.

Quotient rule.

This is the exact sort of uninsightful Bourbakian proof you were deriding a few days ago, though. Neat, but uninsightful.

Also this criticism is really unimpressive to me. Lame. You don't like the simpler and easier to understand route?

Anyway, your idea is very nice too. But analyticity is quite a strong assumption, isn't it?

Most functions aren't even continuous.

Easier to follow, sure. But it tells you nothing about why that's the case.

Yup, most functions that we actively deal with aren't though!

Easier to follow, sure. But it tells you nothing about why that's the case.

also makes sense why it's basically the way to solve differential equations nontrivially,

If you have an extra hypothesis, it's not even the same theorem.

Is the version you gave constructive in some way? Connected with a larger theory that explains how to solve a problem connected with the theorem? Or is it just that Taylor series are more convenient? Just curious.

You're using the royal we like my brain is a math textbook you are writing. I'm not even a mathematician.

Well, as you pointed out, in order to use the quotient rule in the way you do it depends on other theorems. You might be assuming the same things, just unaware of it.

Right, but isn't this still (a version of) the Picard-Lindelof theorem that another commenter brought up?

Well it is a special case of it, but it definitely doesn't NEED such a high-powered theorem to prove. It can be done by the Mean Value Theorem as follows:

Assume that f is such that it is differentiable everywhere with derivative 0. Then by MVT:

[f(a)-f(b)]/(b-a) = f'(c) = 0 for any two points a, b. Thus, f(a) = f(b) so f is constant. Thus only constants have derivative 0.

The general case immediately follows:

Assume f and g are such that f' = g'. Then:

f' - g' = 0 (f-g)' = 0 By what we just proved, f - g = C f = g + C

You don't need analyticity to use the mean value theorem.







https://www.reddit.com/r/math/comments/r3w8r/any_other_functions_than_ex_and_0_that_are/c42zyii/

It's more that, analyticity is a really important concept, and is incredibly useful, so understanding why series expansions work and how they relate and explain basic properties is more important than just understanding the properties themselves.

Even just past e^x as an eigenfunction, power series expansions are absolutely used to solve higher order differential equations, and crop up in all sorts of places.

also makes sense why it's basically the way to solve differential equations nontrivially,

After watching a documentary about Alan Turing, I started thinking about Turing machines again. They’re a very useful tool for computer science proofs and an elegantly simple computational model. Beyond this, however, Turing machines seem relatively useless in terms of “real-world” applications. Any computable function can be computed by a Turing machine, but you probably wouldn’t want to use one as a compilation target: it would be too inefficient.

One possible application for Turing machines is to use them as programs to generate data procedurally. Their simplicity makes it possible, for example, to create working programs randomly. Generating a random stream of x86 instructions that doesn’t crash could be tricky, but with a Turing machine, it’s quite easy. I decided to try something like this to produce so-called generative art. The program I wrote generates random Turing machines that operate on a two-dimensional grid instead of a one-dimensional tape. The symbols written on the grid can be interpreted as colors, and voilà: we have procedural drawings.