For lack of a better word, Modifiers.

[epkid08]

This has to do with the properties of numbers, so I think I posted this in the right place.

My main idea is kind of hard for me to get across. What if we're wrong about our [modern] math? What if, near the bottom of our math basics, we've missed something, something maybe not apparently important now, but will be important in the near future. What if we've been building math for years and years off of a basis that's actually wrongly configured?

I wonder:
Things like squaring a number - $$4^2=4*4$$ - Okay, we have uses for this and so we've concocted notation for it, and have set rules, guidelines, and properties for it. - So basically this modifier (I know there's a better word, just can't think) takes its' number and multiplies its' number by the number itself. - Seems almost complicated. - My idea is that, I can come up with complicated processes too, I can give it a name, I can give it properties as well. - My mind leads me to wonder, if mathematicians have forgotten important modifiers. - Could they lie undiscovered as we speak?

I'm not trying to say that there's something wrong with our current math, but there's possible something missing. - Missing properties, modifiers, rules and the like...

Or possibly, maybe some things are wrong in our modern math:
Let's take negative exponents for example. Who's the one that decided that a number with a negative exponent, $$x^-2$$, equals the reciprocal of the number to the opposite of its original exponent, $$1/x^2$$? Who's to say it shouldn't actual equal $$x^-2=-(x^2)$$? I understand we have many properties that come after this, $$x^-2=1/x^2$$, such as $$x^a * x^b = x^(ab)$$, and uses such as scientific notation, but they were built off of that idea, and what if that idea was wrong?

This topic wasn't made to bash our modern mathematics. The examples I used were hypothetical. I'm just wondering if IT'S POSSIBLE, that we're forgetting something. (also, if someone knows the word that I wanted to use in place of 'modifier', please tell me!)

 

[matt grime]

It's _just_ notation chosen for utility. It's not embodying anything important itself, and has no intrinsic meaning.

 

[epkid08]

I mean we have specific notation for it because it can be applied usefully in many ways. If it didn't have these applications, it wouldn't be important.

 

[HallsofIvy]

There certainly ARE things that we don't know- because we can't know everything.

However, those ideas that we DO know are based on PROOF. Saying that x^-2 is the same as 1/x² is not arbitrary- it's because the basic properties of exponentials will work with that definition and not any other definition. Similarlly with multiplication and squaring- the results are based on theorems that have been PROVED. That's the whole point of mathematics!

 

[CRGreathouse]

I think the points are
1. Are there (simple) functions which have not been explored?
2. Are commonly-used systems of mathematics consistent?

 

[epkid08]

I think the points are
1. Are there (simple) functions which have not been explored?
2. Are commonly-used systems of mathematics consistent?

Yes, that's about it.

 

[Dragonfall]

Well, 2 cannot be answered, but "yes" is a safe bet. And at this point any simple functions which have not been explored probably don't have any intrinsic interest. Unless they are applied to study something concrete and interesting.

 

[CRGreathouse]

Well, 2 cannot be answered, but "yes" is a safe bet. And at this point any simple functions which have not been explored probably don't have any intrinsic interest. Unless they are applied to study something concrete and interesting.

I think that 2 can be addressed. Weaker models have been studied, and there's always a chance for a contradiction to be found -- hopefully in something strong like Tarski's axiom rather than good ol' ZFC.

Speaking of which: does anyone have a good chart of the relative constancy strengths of various systems handy? Like GST < ZF = ZFC = NBG = ZFC + CH < TG?

 

[robert Ihnot]

epkid08: Let's take negative exponents for example. Who's the one that decided that a number with a negative exponent, , equals the reciprocal of the number to the opposite of its original exponent?

Since $$\frac{x^3}{x^2}=x$$ It doesn't take very much to see how well negative exponents work out. Anyway, one call always use induction to prove such things in general.

But we have to come up with the definitions to get started. And, as matt grime points out: It's _just_ notation chosen for utility.

 

[epkid08]

epkid08: Let's take negative exponents for example. Who's the one that decided that a number with a negative exponent, , equals the reciprocal of the number to the opposite of its original exponent?

Since $$\frac{x^3}{x^2}=x$$ It doesn't take very much to see how well negative exponents work out. Anyway, one call always use induction to prove such things in general.

But we have to come up with the definitions to get started. And, as matt grime points out: It's _just_ notation chosen for utility.

Yes, but you have to realize that your example, in turn, was built off of another statement, one that could possibly be wrong.

 

[Dragonfall]

Speaking of which: does anyone have a good chart of the relative constancy strengths of various systems handy? Like GST < ZF = ZFC = NBG = ZFC + CH < TG?

I wish there were one. I can never remember any of these damn relative consistency results. Especially if you go the nonstandard route with non-well-foundedness and whatnot.

 

[CRGreathouse]

I wish there were one. I can never remember any of these damn relative consistency results. Especially if you go the nonstandard route with non-well-foundedness and whatnot.

I'm particularly interested in that side of things. There are standard results (which I can never seem to remember) about certain fragments of ZF (ZF - foundation, ZF - infinity, etc.), but what about stronger theories lacking such parts? What about nonstandard theories with extra axioms or removed axioms?

 

[kenewbie]

Yes, but you have to realize that your example, in turn, was built off of another statement, one that could possibly be wrong.

It important to separate notation from operation from definition.

If we start at the very beginning, by *defining* the natural numbers:

1 is the quantity equal to numbers of you.
2 is the quantity equal to your number of hands.
3 is the quantity equal to the number of bones in your index-finger.
and so on.

Then you *define* the operation of addition to be collecting quantities of natural numbers.

Since the numbers of you combined with the number of your hands is equal to the number of bones in your index-finger, 1 + 2 = 3.

Then there are observations related to addition. You are out collection berries and you notice that one bush has 2, while the other has just 1. It does not matter what order you pick them up in, you always end up with 3 berries. (Unless a bear comes, in which case you don't really care much for math anymore).

So, 1 + 2 = 3 = 2 + 1. The order of addition does not matter.

Then perhaps we introduce some notation

2 + 2 + 2 + 2 = 8 .. this is tiresome to write, so let's just write it like this instead:

4 * 2 = 8

..and so on and so forth. Basic arithmetic is build like this. Which of statements above require proof?

k

 

[matt grime]

Yes, but you have to realize that your example, in turn, was built off of another statement, one that could possibly be wrong.

Which statement could possibly be wrong? The only thing in there was that x^3 divided by x^2 is x. That seems fairly obvious and true to me. As robert says, this suggests a useful way to define x^-1 etc. One that works, and is consistent.

 

[thompson03]

Let's introduce some terminology here:

The distinctions you need to make are between

Definitions : completely arbitrary man made ways of describing something
Axioms (or postulates) Supposedly self evident truths

Theorems: Something Proven from Definitions and Axioms

Godel proved that not all theorems can be proven, or even known for a complicated enough system

 

[matt grime]

Axioms are not 'self evident truths'. There was a long and tedious thread on this in general maths very recently.

 

[epkid08]

Oh yea, I forgot to add this, some examples of what I mean are the trig functions, mod function, etc; Also, not just functions like that, but operators too, i.e. 4@4=44.