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Is 1 Prime, and Does It Matter?

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Re: Is 1 Prime, and Does It Matter?

#131

Post by AStonesThrow »

I've been fascinated by numbers lately, and one of my go-to tools is a simple mobile app that calculates all the divisors of a given number. So I can determine prime numbers, and readily factor the non-primes. And it's been eye-opening.

Now I'm no crackpot numerologist, adding up the numerical values of Bill Gates' name, or telling you who shot JFK. But I can tell you that the main launch pad 39A at Cape Kennedy was not numbered by accident -- look it up in the Book of Psalms. And it's interesting how the city buses around here are numbered. For example, the 68xx series; I look up Psalm 68 and I can definitely imagine the bus singing that as it lumbers down the road -- can't you?

Back to primes -- if we consider the top numbers authorities of our times, such as the US Post Office, city planners, and the telephone company (circa 1970s). I ran a chunk of ZIP codes from Southern California and discovered that some are the factors of two quite large prime numbers. Others yield interesting factors. Once again I pull out my Book of Psalms.

There are plenty of other "hermeneutics" to interpret assigned numbers, especially street addresses. And as for phone numbers, I've gone back to figuring out "what do they spell" on a standard TouchTone keypad, because sometimes it's quite informative.

It's no accident, for example, that the hospital where I was born is located at 4077 5th Avenue. And that number assigned by city planners, many decades before M*A*S*H was written or went on TV. Significant nonetheless.

I also figured out a few prime numbers related to my own life, and others that are recurring tropes, just cropping up at interesting times. What's your social security number? Have you sort of broken it down and pondered if those numbers turned up again and again in your life? Every time I see a number now, I'm compulsively factoring it out in my head. Is it prime? It feels prime. I'll check it in the app later; try some mental math for now.

I'm also counting things more often now. How many spokes in a wheel? How many petals in a flower, especially a flower depicted in art. How many brick courses in that interesting wall they built? Plug any interesting numbers back into the divisors app. Finding the primes, find the factors, just ponder numeric coincidences. It's fun. So many signs and signals, hidden in plain sight before us. Buses singing Psalm 68 as they take on passengers. Launch pads singing Psalm 39 as Europa Clipper slips the surly bonds of Earth. What's on your telephone dial?

Re: Is 1 Prime, and Does It Matter?

#132

Post by ForOldHack »

pinkmuffinere wrote: post #49
gerdesj wrote: post #45

Earlier quoted context omitted.

You might start here: https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach That's the GEB I mentioned above.

I know this is a great book, it’s been on my to-read list for about 5 years. But I never get to it. Is there not another (shorter) discussion I could read on this? Even an academic paper would be acceptable.

You could read Kurt Godels paper,but it's literally undecyperable. The book is one on the best reads ever. It will also teach you how to think in very very formal ways. It made Calculus half the class it was, and I breezed through finite math.

Propositional Calulus will teach you to think in symbols you cannot even fathom. This alone is worth every minute reading the book.

Every few years I reread it, and get a new sense of solving problems. The book can be divided into parts... But the whole...

Re: Is 1 Prime, and Does It Matter?

#133

Post by ForOldHack »

mathgeek wrote: post #77
murderfs wrote: post #48

Earlier quoted context omitted.

A good example of this is the natural numbers. Algebraists usually consider zero to be a natural number because otherwise, it's not a monoid and set theorists want zero because it's the size of the empty set. My number theory textbook defined natural numbers as positive integers, but I'm not entirely sure why.

> My number theory textbook defined natural numbers as positive integers, but I'm not entirely sure why. Since both the inclusion and exclusion of zero are accepted definitions depending on who’s asking, books usually just pick one or define two sets (commonly denoted as N_0 and N_1). Different topics benefit from using one set over the other, as well as having to deal with division by zero, etc. Number theory tends…

Number theory includes zero as the identity element for addition, much as 1 is the identity element for multiplication.

I am totally assuming you knew this already.

Re: Is 1 Prime, and Does It Matter?

#134

Post by [deleted] »

Maxatar wrote: post #88
sam_ezeh wrote: post #87

Earlier quoted context omitted.

>Any definition of the natural numbers will also define things that look very similar to natural numbers but are not actually natural numbers This isn't correct. This is only true for first-order theories of the natural numbers using the axiom schema of induction. Second-order Peano arithmetic with the full axiom of induction has the natural numbers as its only model. This property is called "categoricity" and you ca…

This isn't correct. While it's true that in second order logic the natural numbers admit categoricity, second order logic lacks axiomatic semantics. So yes, there is a single set which can be called the natural numbers in second order logic (namely the intersection of all sets that satisfy Peano's axioms), but this set has no interpretation. You can adopt Henkin semantics to give the naturals an interpretation, which…

[deleted]

Re: Is 1 Prime, and Does It Matter?

#135

Post by ForOldHack »

margalabargala wrote: post #99
munchler wrote: post #18

Other good nerd-sniping math questions: 0^0 = 1? Yes, it’s simpler that way. 0! = 1? Yes, it’s simpler that way. 0/0 = ∞? No, it’s undefined. 0.9999… = 1? Yes, it’s just two ways of expressing the same number. 1+2+3+… = -1/12? No, but if it did have a finite value, that’s what it would be.

> 1+2+3+… = -1/12? No, but if it did have a finite value, that’s what it would be. The other ones, sure, but I'm not following this one.

You missed the lecture on the missuse of infinities.

If I have inf*k = inf,and dvide both sides by inf... ( The misuse) Then 1 = any K including 1/12. Now this is useless in calculus and number theory, but in quantium field theory it is a useful tool.

So inf = 1/12 and a non convergent series = a constant, but you have misused dividing infinity by itself to get it.

Infinity for division? It's useful, like counting chickens starting at zero. L'Hoptals rule is a very useful tool, but do not misuse it.

Re: Is 1 Prime, and Does It Matter?

#136

Post by drdeca »

gerdesj wrote: post #27
reaperman wrote: post #6

Earlier quoted context omitted.

Yes, it's more of a convention where we assume language like "...ignoring the trivial case of 1 being an obvious factor of every integer." It's not interesting or meaningful, so we ignore it for most cases.

I'm no expert but: "...ignoring the trivial case of 1 being an obvious factor of every integer." I remember quite a big chunk of GEB formally defining how integers are really not trivial! The main problem seems to be is that you soon end up with circular reasoning if you are not razor sharp with your definitions. That's just in an explainer book 8) Then you have to define what factor means ...

The previous poster didn’t describe the natural numbers as trivial. Rather, described a case as trivial.

Specifically, the case of the divisor being 1.

Re: Is 1 Prime, and Does It Matter?

#137

Post by gblargg »

Spivak wrote: post #101
scoofy wrote: post #92

Just a note from your friendly philosophy degree holder: Axioms are arbitrary. Use the axioms that are the most useful.

Definitions are neither true nor false. They're either useful or not useful. The question of whether or not the integer 1 is a prime doesn't make sense. The question is is it useful to define it as such and the answer is a resounding no.

Agreed. Definitions are made to differentiate things in a way useful for some goal. The question "Is X an M?" without a context or goal basically picks up whatever vague goals or purposes a person has lingering below the surface of consciousness, differing from what other participants have below theirs, leading to different answers, with no way to select the best one. In the case of what is considered prime, it's a matter of what definition simplifies the things that use it. It could be that two concepts are better, one including 1 and the other not including it. Since it's just a language shorthand, it makes no fundamental difference other than efficiency and clarity in communication about math.

Re: Is 1 Prime, and Does It Matter?

#138

Post by pja »

btilly wrote: post #59
pja wrote: post #51

Earlier quoted context omitted.

> Correct, it's impossible to specifically and formally define the natural numbers so that addition and multiplication work. Any definition of the natural numbers will also define things that look very similar to natural numbers but are not actually natural numbers. Are such objects not inevitably isomorphic to the natural numbers? Can you give an example of a formal definition that leads to something that obviously…

The Peano Axioms lead to both the standard model of arithmetic (the integers that we want), and nonstandard models. See https://en.wikipedia.org/wiki/Non-standard_model_of_arithmet... . In that article you'll see references to "first order logic" and "second order logic". First order logic captures any possible finite chain of reasoning. Second order logic allows us to take logical steps that would require a potentia…

All of these models appear to contain infinitely sized objects that are explicitly named / manipulable within the model, which makes them extensions of the Peano numbers though, or else they add other, extra axioms to the Peano model.

If you (for example) extend Peano numbers with extra axioms that state things like “hey, here are some hyperreals” or “this Goedel sentence is explicitly defined to be true (or false)” it’s unsurprising that you can end up in some weird places.

Re: Is 1 Prime, and Does It Matter?

#140

Post by chongli »

JadeNB wrote: post #115
chongli wrote: post #95

Earlier quoted context omitted.

I don’t see why it’s a problem that the empty set cannot be a group. The empty set, being empty, lacks an identity element. Thus all groups are non-empty. The same is true for any structure which posits the existence of some element. Of course it cannot be the empty set.

> I don’t see why it’s a problem that the empty set cannot be a group. The empty set, being empty, lacks an identity element. Thus all groups are non-empty. It's not necessarily a problem that the empty set cannot be a group. (Although the only reason that it cannot is a definition, and, similarly, the definition of a field requires two distinct elements, which hasn't stopped some people from positing that it is a pr…

If you don’t allow the empty set to be a set then you break the basic operations of set theory. For example, to show two sets are disjoint you compare their intersection with the empty set.

In an alternative axiomatization (without the empty set) you’re going to need to create some special element which belongs to every set and then your definition of disjoint sets is that their intersection is equal to the trivial set containing only the special element. What a clumsy hack that would be!

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