MATHEMATICS

 

Ultimately while Minsky was not wrong in saying, in an early textbook, that programming is not mathematics, and Dijkstra was not wrong in saying, in an early speech, that it's all basically mathematics, the truth is somewhere in between.

I wrote earlier of the distinction between mathematics and physics.

What is mathematics about?
 
Mathematics is essentially about possible measurements. The criterion, after all, is consistency.
 
So physics is about natural existence — what's actually measured by observers. (DeWitt, Clifford Truesdell, John Wheeler, Dirac have essays on the subject.) Nature being any subset of all that exists - the universe.
 
That meanwhile indeed is the reason Dirac's favorite approach of physical interpretation of mathematics works well or at all: what is not measured does not exist. (It has no behavior — does not induce a dynamic on any other system in the universe.)
 
So the so called unreasonable effectiveness of pure — invariant to physical realizational baggage — consistent speculation is simply that mathematics is a theory of measurement operations as they may be combined; physics is a theory of existence.
 
Of course they're intimately related — more than by random coincidence. Yet not the same. What can be measured is not necessarily measured. It's merely possible. Existence is more than that. — But what cannot be measured does not exist.
 
That beats random guessing about what's out there in the world, — whatever else is there. So this is therefore far more likely than random guessing at being correlated the truth — which is a map of things that do exist.
 
But indeed while all physical propositions are therefore a proper subset of all mathematical propositions, physics is not a proper subset of mathematics, not a part of mathematics, but mathematics is an important tool, one among many in a toolbox, in physics and physical reasoning.
 
For example, how much of pure mathematics — say http://www.math.harvard.edu/~lurie/papers/HA.pdf — is physically meaningful? None of was intended to be meaningful. (It's a good thing that intention is not always required to be right, rather (paraphrasing David Hume) only coincidence.)
 
When you stop to think about it, possibly quite a lot. The joke in mathematics is this: — we'll see later. So how much of pure mathematics is physically meaningful? Not all — but a lot.
 
Essentially the subject would be measurements where information is not fundamentally lost, even when coarse measurements, ones at higher granularity, are performed and reasoned about.

Now rather I'd like to add to that.

Agreeing more with Wolfram and somewhat contrary Galilei, it's not even so much that we must learn learn mathematics to understand nature because nature is written in the language mathematics.

Nature maybe is written in mathematics; true. For that's the language of possible measurement. And nature basically is measurement. One thing creating a dynamic on another. But we often read books written in another language in translation.

If that were the case, possibly we'd not notice it regarding nature however.

There is a more pragmatic reason why mathematics is important.

You can learn and acquire and have knowledge. It's not so much the language of nature as the language of science.

Because it's the only thing that allows you to work on little pieces independently. — Then fit them together. — Such that they actually would fit together. — Which doesn't happen for ordinary language. — Only for mathematical language.
 
This is another side effect of making a language based around consistency.—Based around abstraction and formal coherence. One thing is independent of another. Yet they do fit if attempted to be put together. — If made into one.
 
We may see that some things must happen quickly if they are to happen. Meanwhile other things must happen slowly if they are to happen.
 
The probability of solving a thousand simple problems correctly is much, much higher than solving one problem a thousand times as complicated or complex. Especially if guessing answers or parts of answers and intelligently proceeding from the guess by a gradient. How brains do work. And how artificial intelligence is built.
 
And the probability of throwing a knife in a target from a foot away a thousand times consecutively is significantly higher than hitting the same target with the same knife once or more in a thousand throws from a thousand feet away.
 
If you can't break complicated and complex things into atoms, it's very difficult and unlikely to get much done. Maybe it can get done. But not much. And most probably not very quickly.

NOTATION

 
This is a blog. — Not a full write up.

I had written a few things about notation earlier.

Related to the topic above, and continuing from there. —

(i) What do you guys think of notation like the following?

E.g., S(#1=1,#1=n,#2=0,#2=N)(#1 ' 2 + #2) for the sum for x^2 + y with x going from 1 to n and y from 1 to N.

(ii) Meanwhile, for example, S*(type_Int, length_10, lowerbound_500) for the dual, a random sequence of 10 integers x, such that any integer in the sequence > 500.

Then we can do something like: S*(type_Int,length_10,lowerbound_500).s(1), i.e., name_A,type_Int,length_10,lowerbound_500.S*.(1.s) — where we generate the list, select 1 and mark it. So we go from {} ↦ A_750,,A_2501,,...,,A_marked_678,,...,,A_15691 ↦ A_750,,A_2501,,...,,A_678,,...,,A_15691.

Also we'll do most things, I suggest, with > rather than ≥. (i) That's more stable. Less probability of oscillation, etc. (ii) Even for discrete variables ordinary calculus max/min can be done for systems of step functions for discrete valuables. Using distributions. Simplifies things.

ABOUT ME

I'm a scientist who writes science fiction under various names.

The magazines that I most recommend: Magazine of fantasy and science fiction, Compelling science fiction, Writers of the future, . . .

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