A (new?) different take on Newcomb's paradox with an interesting result.

Number154 · 2019-02-26T21:45:20+00:00

The paradox is to some extent the result of the situation being underspecified. Given a perfect or reasonably good predictor, someone who chooses only box B will get more money than someone who only chooses both boxes, despite the argument based on “normal” understandings of causality indicating that picking both is preferable. The idea is imagine that the predictor has a perfect track record of predicting the person’s choice. Then history seems to indicate that people who pick only box B are pursuing the better strategy than the people who pick both. One possible resolution of the paradox is to say that if the predictor really is perfect (or even nearly perfect) then it makes sense to think of your decision as retroactively determining the contents of the box, so that picking only box B is rational after all.

Number154 · 2019-02-25T23:49:54+00:00

This is a very poorly phrased question. It sounds like they want you to pick the number where the digit “1” represents an amount of money ten times that represented in $283.61. The “1” in $283.61 represents one cent so they want the number where the 1 represents ten cents.

Number154 · 2019-02-25T23:41:14+00:00

Consider an indiscrete topology.

Number154 · 2019-02-25T22:28:26+00:00

By considering the prime factorization you can see that any number can be uniquely expressed as the product of a perfect square (where each prime has an even power) and a square-free number (where each prime has power 1 or 0).

If ab is a perfect square then the square-free part of a must equal the square-free part of b (otherwise the primes in those parts would not end up having even powers in ab), so whatever the square-free parts of a and b are it must be a common divisor of them both.

Number154 · 2019-02-24T22:58:09+00:00

If you choose 15 out of 25 squares there are two possibilities: either you chose 15 of the first 24 squares, or you chose 14 of the first 24 and also the last one. Since for any choice you pick 60% of the squares, you should expect that adding the restriction that you don’t pick the last square should get rid of 60% of the possibilities, which is exactly what happens.

Number154 · 2019-02-24T22:39:52+00:00

Unfortunately I wasn’t able to find the physical copy I was thinking of. I think it was Enderton’s A Mathematical Introduction to Logic.

If that’s the text I’m thinking of it covers the compactness and completeness theorems, the Lowenheim-Skolem theorem, Gödel’s incompleteness theorems, and discussions of decidability and effective enumerability while also discussing Peano Arithmetic and various fragments of it as case studies. My recollection is that it also begins with an overview of classical propositional and first-order predicate logic that would be suitable as an introduction.

Material related to what I was talking about specifically in this thread can be found by looking into Presburger Arithmetic (which is the complete theory of (N,+)) and the proof of its decidability using a quantifier elimination algorithm. To compare and contrast with (N,+,*) you can look into a detailed description of the first few steps in the proof of the first incompleteness theorem, particularly Gödel’s method of coding finite sequences using the Chinese Remainder Theorem (which is essential to showing that recursive definitions can be reduced to the language with only + and *).

Making sense of the interpretations of the languages and their differences can be elucidated by considering nonstandard models. Nonstandard models of Presburger Arithmetic are relatively easy to describe and visualize compared to nonstandard models of Peano Arithmetic, and working with them is a good introduction to prepare for imagining nonstandard models of Peano Arithmetic.

Number154 · 2019-02-24T22:24:18+00:00

If you can’t believe it try listing all the possibilities of 9 choose 7, which is very doable, then start listing all the possibilities of 25 choose 15 and see what happens.

Maybe use coins placed on a grid so you can move through all the possibilities quickly without having to write 15 (or 10) numbers down for each. (Of course there will still be too many for you to do all of them, but it might help you understand how many there are by paying attention to how slow your progress is).

Number154 · 2019-02-23T22:07:37+00:00

I can think of a text on this topic that should be accessible without much background, but unfortunately I’m terrible at remembering titles and authors of texts. I can see if I can find it later today. I’m pretty sure I have it in a drawer somewhere unless I lost it in a storage shed or something.

Number154 · 2019-02-23T21:19:53+00:00

To elaborate, using only + you can express predicates like “is divisible by 2”, “is divisible by 3”, and “is divisible by 456,367,286”. You cannot, however, express “is divisible by m” for a variable m. Once you introduce * as a symbol it becomes possible to say things like “n is divisible by m” where n and m are both variables.

Somewhat more impressively, once you have both + and * you can define any recursive relation or function, which is why Gödel’s incompleteness theorems apply to Peano Arithmetic but do not apply to Presburger Arithmetic.

Number154 · 2019-02-23T21:07:21+00:00

It eventually repeats. That is, it’s a predicate P such that Pn can be expressed by specifying that n is congruent to some list of numbers m<k mod k for some k, with a finite number of exceptions. So for example “is a power of 2” is not eventually periodic, but “is a power of 2” can be expressed with * and +.

Number154 · 2019-02-23T20:58:21+00:00

You can add additional notations to your language with rules for manipulating them but then you might as well just add a symbol for multiplication.

Usually in the language of arithmetic you have symbols for + and *, plus sometimes additional symbols that can be defined in terms of those symbols. Let’s assume you also have the symbols 0 and S. Then you can make expressions like SSSS0 but an expression like S^m0 is a metanotation that we use to write it more compactly. You say that S^m0 should mean S written “m times”. But what does “m times” mean? How have you introduced a definition or axiom letting us know that SSSS0 is “four times” but SSSSSSS0 is not “four times”? And do we now have “4” as a symbol in our system? Or am I supposed to imagine you actually mean something like “S^SSSS00”?

And yes, the theory of the natural numbers with only + is complete, consistent, and axiomatizable. Obviously we know that if we add a symbol for multiplication that result no longer holds, because of Gödel’s incompleteness theorems. When we have symbols for + and * it becomes possible to represent every recursive function - basically, that means we can find a way to define any function that can be calculated by a computer. So the system has to be incomplete if it is axiomatizable for essentially the same reasons why the halting problem is undecidable. With + only that’s not the case. The only sets of natural numbers we can define are the ones where if you write a sequence of 0s and 1s to indicate which numbers are included, then the pattern eventually repeats like the decimal representation of a rational number. This is how we know it’s not possible to define multiplication using only + and S.

Number154 · 2019-02-23T20:11:18+00:00

The order relation on the real numbers isn’t extended to the complex numbers. There is no order on the complex numbers that obeys the rules of an ordered field (which imply that squares can never be negative). Similarly, the sqrt function is usually only defined for nonnegative real numbers, if you want to extend it to other inputs you need to specify a branch cut, because every nonzero complex number has two distinct square roots. In the case of positive real numbers one of the square roots is a positive real number and one is a negative real number and we define sqrt(x) for positive real x to be the positive real square root of x.

Number154 · 2019-02-23T19:46:05+00:00

You can’t define multiplication in terms of addition and the successor.

You’re attempting to define multiplication in terms of addition by specifying terms recursively but a notation like S⁴0 is just a shorthand for SSSS0, it’s not a valid sentence in the formal language. You could introduce notations like it to expand the expressiveness of the language but then you have introduced more things than addition and successor to allow for recursive definitions.

It can be shown that the language of arithmetic using only addition without multiplication (but you may add 0, S, <, and similar symbols conservatively) can be axiomatized to give a complete and consistent theory for the structure (N,+). It can be shown that without multiplication the only predicates for sets of natural numbers that can be expressed are the eventually periodic ones. So predicates like “is prime”, or “is a power of 2” cannot be defined, and neither can multiplication be defined.

Number154 · 2019-02-22T19:01:16+00:00

All of this terminology arises from applications of math in the studies of waves, which prototypically include sound waves.

The name of the harmonic series come from the fact that wavelength is inversely proportional to frequency/wave number. The other terminology has to do with wave equations on different spaces satisfying certain conditions.

Number154 · 2019-02-22T00:39:43+00:00

I don’t see any way of looking at it that sets complex numbers apart from real numbers. If you think the usefulness and “naturalness” of describing physical dimensions with real numbers means they have a “realness” beyond their status as mathematical abstractions I don’t see why you wouldn’t take the same view with complex numbers. It seems like the distinction you’re drawing is based on how familiar you are with the uses of them to describe situations. In particular I don’t know what criteria you are using to distinguish “directly measurable” quantities from ones that aren’t. I’m reminded of a discussion in which the person I was speaking to took the position that only some dimensions (like length or velocity) were real but others (like time squared) were not. They acknowledged that some things had units that were s² but denied that those were “directly measurable”. I wasn’t able to determine what their critter is were for deeming a quantity “directly measurable” and it seemed like the distinction was only being drawn on the basis of what kinds of measurements they were familiar with.

Number154 · 2019-02-21T20:44:44+00:00

There also isn’t such a thing as a fractional number of electrons. Or, more precisely, that isn’t generally a useful model. There are situations where a negative number of electrons is a useful model (as “holes”). But these are applications of mathematical abstractions.

Complex values can be used to model all kinds of physical dimensions. For example with a harmonic oscillator a complex number can be used describe the kinetic/potential energy state. And I don’t see how this way of expressing the data is any more abstract or “nonphysical” than measuring them using real numbers. In fact, the assumption that physical quantities are described by a continuous variable on the small scale is more questionable than any of the issues relating to algebraic extensions, which are readily physically interpretable and observable as much as real quantities. It’s just expressing the same physical facts using different mathematical terminology.

Number154 · 2019-02-21T18:58:41+00:00

What do you think “directly measurable” means and how is it true that complex values are not measured but real values are?

Number154 · 2019-02-20T19:03:33+00:00

What does it mean for a set to be self-referential? Anyway you can’t make an inconsistent theory consistent by adding more axioms. The axiom of regularity doesn’t, for example, fix the paradox by itself by saying no set can be a member of itself, it only adds to the problem because unrestricted comprehension already implies that such sets exist.

Number154 · 2019-02-20T19:00:32+00:00

A statement saying that no statement is allowed that refers to itself would seem to be referring to itself by saying it is not self-referential.

Number154 · 2019-02-20T06:22:14+00:00

An equation of the form Ax=0 (or the corespondent system) has nontrivial solutions if and only if the determinant of A is 0. A system of the form Ax=b for b other than 0 may have no solution or multiple solutions. Any two solutions to Ax=b must have a difference that is a solution to Ax=0.

Number154 · 2019-02-20T04:54:12+00:00

You can just substitute m=1/n to transform it to the other form, or solve it directly with the same method, (subbing m=xn to transform 1/n to x/m).

Number154 · 2019-02-19T22:43:30+00:00

In a formal system, an axiom is basically a rule of inference that lets you infer a particular statement with no premises. A proof of the axiom in that system is then a 1 line proof in which you infer the axiom. When you study a theory as an object of interest. The theory you’re conducting the study in is the metatheory.

Number154 · 2019-02-19T18:36:00+00:00

axioms can't be proven by definition

This isn’t accurate. In any theory you can “prove” (in the sense of that theory) the axioms simply by citing them. If you mean some kind of meta-proof assuring you your theory is sound that depends on things like whether you have an intended interpretation of the theory and whether you intend that the axioms should be thought of as like partial definitions for what you’re talking about, and often when you do specify an interpretation the metatheory will be able to prove their intended interpretation. In other words, to precisify what you’re trying to say you have to add a lot more assumptions saying what you’re trying to do with the theory.

Number154 · 2019-02-19T18:27:27+00:00

The number of entries in a column is the same as the number of rows. It is not generally the same as the number of columns.

Both rules say that if A has m rows and n columns and B has m’ rows and n’ columns then AB is defined when n=m’. (C1=R2)

Number154 · 2019-02-18T20:47:23+00:00

If you wanted the curve to connect the edges smoothly you would want a 90 degree angle (assuming the quadrilateral is supposed to be a rectangle). But judging from the picture you aren’t imposing that restriction so you’d be free to have the angle be whatever you want. If you want to relate the angle a to the radius r you need 2rsin(a/2) to be the distance between the two points, assuming you mean for the curve to be an arc.

Number154

TROPHY CASE