Errol Morris Refutes It Thus • Hacker News Board

Re: Errol Morris Refutes It Thus

#31

Post by romwell » Wed, Jun 06, 2018, 8:54 PM UTC

fjsolwmv wrote: post #24

romwell wrote: post #22

Earlier quoted context omitted.

>An interesting example that someone gave from elementary mathematics is the derivation of the quadratic formula... A tangential point: this is indeed a horrible way to derive the formula, for the reasons you mentioned. If anyone is curious, here's a better way to think about it. The graph of ax² + bx + c is just the graph of ax² translated. Keeping that in mind, let's investigate. First, consider a very easy problem…

And that's just a special case of Completing the Square (but has a very handy visualization). https://en.m.wikipedia.org/wiki/Completing_the_square

Not quite. "Completing the square" is an algebraic step; the argument is geometric (translation of the parabola). One can learn how to complete the square (symbolic manipulation) and be completely unaware of the geometry.

If you never learned the algebraic trick, would you invent it when solving this problem? It's not immediately clear from the algebra that a quadratic polynomial ax^2 + bx + c even can be rewritten in the form a(x-Q)^2 - R.

Conversely, while this technique involves algebra that amounts to completing the square, it exhibits a general technique.

If you want to say "..and that's just a special case of..", homotopy continuation methods[1][2] -

- because the idea here is transforming the simpler polynomial x^2 - C into a more complicated one, and seeing what happens to the roots.

[1]https://en.wikipedia.org/wiki/Numerical_continuation

[2] https://en.wikipedia.org/wiki/Numerical_algebraic_geometry#H...

Re: Errol Morris Refutes It Thus

#32

Post by romwell » Wed, Jun 06, 2018, 9:03 PM UTC

jacobolus wrote: post #25

romwell wrote: post #22

Earlier quoted context omitted.

>An interesting example that someone gave from elementary mathematics is the derivation of the quadratic formula... A tangential point: this is indeed a horrible way to derive the formula, for the reasons you mentioned. If anyone is curious, here's a better way to think about it. The graph of ax² + bx + c is just the graph of ax² translated. Keeping that in mind, let's investigate. First, consider a very easy problem…

Personally I prefer to break things into three steps: First put the equation into the form x ² = 2 ax + b Now complete the square: ( x – a )² = a ² + b Finally, x = a ± √( a ² + b)

Sure, it's short. But it doesn't get to why one would do these steps if you don't know they are going to lead to a solution. Completing the square is a non-obvious step to make.

The argument in my previous comment attempts to provide motivation for such a step, starting from simpler questions. It uses the geometry of the problem, and builds up from solving a simpler problem first.

That approach also uses the notion of transformation and invariance (seeing what happens to the roots when we move the graph around, and noticing that the distance between the roots doesn't change we shift horizontally).

Again, the important part here, is that you could lead someone to ask the same questions and have them answer them themselves.

"Let's solve this equation. Looks complicated. Can we solve a simpler problem first? What would a simpler problem be? Now how can we make it a little more complicated, and how does it affect the answer?".

And that's how math is done.

After all is said and done, one can extract "completing the square" as a shortcut technique. But that's what it is - a shortcut through the woods. Learning a shortcut won't teach you how to walk in the forest on your own.

Re: Errol Morris Refutes It Thus

#33

Post by romwell » Wed, Jun 06, 2018, 9:05 PM UTC

schoen wrote: post #23

romwell wrote: post #22

Earlier quoted context omitted.

>An interesting example that someone gave from elementary mathematics is the derivation of the quadratic formula... A tangential point: this is indeed a horrible way to derive the formula, for the reasons you mentioned. If anyone is curious, here's a better way to think about it. The graph of ax² + bx + c is just the graph of ax² translated. Keeping that in mind, let's investigate. First, consider a very easy problem…

Thanks! It's quite possible that my example came from Paul Lockhart or a proponent of somewhat similar ideas.

Indeed, and I hope these ideas gain more ground. Mathematics as a magic trick needs to end.

Re: Errol Morris Refutes It Thus

#34

Post by ShabbosGoy » Wed, Jun 06, 2018, 9:52 PM UTC

mannykannot wrote: post #30

ShabbosGoy wrote: post #6

Earlier quoted context omitted.

> I agree with Weinberg that Aristotle understood physics worse than many schoolchildren do today. Which is not to say that Aristotle was stupid, but just that your knowledge is a function of the times you live in and the volume of past human thought you've had the privilege to learn from. Would physics as a discipline exist in its current form without thinkers like Aristotle? That’s the more important question for m…

The course of science is shaped by the natural world at least as much by people. It needs people with the curiosity of Aristotle, but how it proceeds depends also on what nature reveals to them, and on what they can pry from nature by using what they have learned so far. My guess is that something like our physics has emerged (or will, relatively shortly) on every world populated with intelligent beings.

> My guess is that something like our physics has emerged (or will, relatively shortly) on every world populated with intelligent beings.

The anthropic principle states as much. I like that you used “intelligent beings” instead of humans, because I could imagine a scenario where synthetic or artificially intelligent beings would arrive to the same conclusions, upon making the same observations of their natural world.

Re: Errol Morris Refutes It Thus

#35

Post by woodandsteel » Wed, Jun 06, 2018, 11:51 PM UTC

neuxenian wrote: post #13

woodandsteel wrote: post #12

Whether or not Kuhn was a radical relativist, a great many intellectuals today are. Radical relativism rests on Cartesian dualism. This is a philosophical doctrine that, as anyone who has studied philosophy knows, is highly problematic. In the last century or so it has been rejected by a long string of major philosophers, include Husserl, Heidegger, Merleau-Ponty, Ricoeur, Whitehead, the later Wittgenstein, Strawson,…

The relativist position in modern philosophy of science has little or nothing to do with Cartesian dualism; it has more in common with the incompleteness theorem in certain respects. The short version is that any theory of observations (i.e., scientific theory) necessarily defines the observational process and therefore the observations themselves, so that it is impossible to separate an explanation from its explanan…

>The short version is that any theory of observations (i.e., scientific theory) necessarily defines the observational process and therefore the observations themselves, so that it is impossible to separate an explanation from its explanandum.

What I am saying is that all the different scientific theories are ultimately based on a universal understanding of reality that is in turn due to the universal nature of the human organism and its engagements with the real world. That's why I included the last paragraph in my comment.

Re: Errol Morris Refutes It Thus

#36

Post by jhbadger » Thu, Jun 07, 2018, 10:59 AM UTC

neuxenian wrote: post #13

woodandsteel wrote: post #12

Whether or not Kuhn was a radical relativist, a great many intellectuals today are. Radical relativism rests on Cartesian dualism. This is a philosophical doctrine that, as anyone who has studied philosophy knows, is highly problematic. In the last century or so it has been rejected by a long string of major philosophers, include Husserl, Heidegger, Merleau-Ponty, Ricoeur, Whitehead, the later Wittgenstein, Strawson,…

The relativist position in modern philosophy of science has little or nothing to do with Cartesian dualism; it has more in common with the incompleteness theorem in certain respects. The short version is that any theory of observations (i.e., scientific theory) necessarily defines the observational process and therefore the observations themselves, so that it is impossible to separate an explanation from its explanan…

But this is ignoring what is is probably the most harmful influence of Kuhn -- the whole idea of "paradigm shifts". For a while nearly every new finding was hyped as a "new paradigm" thanks to Kuhn's SSR -- in the field of statistical genetics, Joe Felsenstein used to joke that he owed his success to the fact that he was the only one doing boring "normal science" in the 1970s rather than trying to shift paradigms.