Terrence Howard on Reality

[Yimpa]

27 minutes ago, Leo Gura said:

Actually, you do.

I’ll talk to an enlightened master who loves pizza then 

[Ero]

7 hours ago, Leo Gura said:

I have a controversial take on the matter. Human mathematics is actually not grounded in axioms. The axioms are a post-hoc rationalization.

As someone who studies graduate mathematics at a top university, I actually agree with this. The axiomatic approach was created for the sake of consistency and it is already leading to problems. My intuition is it represents a Blue-stage conception of mathematics as a formal logical system, something Gödel already disproved. That said, the braindead stuff Terence Howard espouses is Purple-level mythology that is entirely disconnected from that precise naturalistic origin of mathematics - as an instrument of abstraction to reason about reality.

P.S. The dichotomy of “realness” of mathematics vs physics is one I have found very good to contemplate about in general. Hint: both occur in Consciousness.

Edited June 17 by Ero

[Ero]

15 hours ago, Hojo said:

@zurewThe axiom that specifically states why 1x1 must = 1

TLDR: It is called the axiom of multiplicative identity for rings in Abstract Algebra when applied to the set of Whole numbers (Z)

Whole numbers as a set defined with an operation of addition, constitute a group. The axioms of a group are:
1. Associativity, i.e the order doesn't matter : (2+ 3) + 4 =9= 2+  (3 + 4) 
2. Identity, i.e an element (0) with which the operation by definition leaves the other element the same: 0 + 2 = 2 = 2 + 0
3. Inverse, i.e an element that negates the other, s.t the result is the identity : 2 + (-2) = 0 = (-2) + 2

In this case the whole numbers are abelian (commutative), i.e the order of summation doesn't matter: 2+ 3 = 3 + 2

When you add multiplication, the commutative group now becomes a ring when you have two extra axioms fulfilled: associativity of multiplication (equivalent to (1), just switch + with *) and distributive law, i.e (a + b) * c = a* c + b*c

A ring can have a multiplicative identity, which in the case of the whole numbers is 1, because by definition when you multiply 1 with anything, you still get the same element: 1x2 = 2, 1x3 = 3, 1xa = a, so just replace "a" with 1 and you have 1x1 = 1. 
 

 

[zurew]

5 hours ago, Leo Gura said:

I have a controversial take on the matter.

Human mathematics is actually not grounded in axioms. It is grounded in the empirical workings of material reality. The axioms are a post-hoc rationalization.

Yeah that is a controversial one, because as far as I understand, mathematicians would say that math would be true regardless how the physical world works. This kind of goes back to modal logic - where we talk about certain things being possible in all  possible worlds (possible here means logical possibility and the set of all physical possibility is inside the set of logical possibility - in other words, there are much more logically possible things than there are physically possible things).

5 hours ago, Leo Gura said:

If you added up two chicken eggs and they amounted to eleven eggs, then our math would be 1+1=11. But this dream doesn't work that way.

I personally disagree with the idea that math is empirically grounded (meaning the truth value of math propositions are not directly depended on the physical laws of this universe) . Like if we would want to go down that route, there is so much ambiguity that could be applied without the usage of extremely strict mathematical definitions (especially defining very strictly what we mean by identity). 

For example, I could empirically justify almost anything that would contradict basic math axioms. Like, If I would want to get silly about it, I could say 1+1=3, because empicially if two people fuck, one children will be born (and even this is not necessarily true, because sometimes 1+1=4 or more depending on what we are talking about twins, triplets or anything else) - I could even add here that 1+1=1 or 1+1=0, because when two people have sex and when one of them has a deadly STD, the other could catch that STD and then both of them could die later down the road. - and here we only talked about human sex, if we go to animals even more crazy empirical cases could be brought up that would almost all contradict basic math axioms.

I can come up with other silly examples if needed, but the basic point here is that that crux of the issue seem to be how strict our definitions are and this doesn't seem to be about anything empirical.

Also where did you empirically observe that 1million*1million=one trillion?  -  you accept a bunch of mathematical statements that you have never observed or empirically verified in the physical universe ever before - hence why this goes back to definitions imo and to inference rules that we all agree on, regardless whats going on empirically .

Edited June 17 by zurew

[zurew]

@Ero Hi , cool that you are here,  you can clean this thread up and correct whatever bs we are talking here

[Salvijus]

What makes no sense in math is that x^0=1. It's just completely random lol. 

Although there was one mathematician here who explained how mathematicians are doing crazy mathematical gymnastics to make it so. Unfortunately I forgot his proof, haha. But it was pretty interesting.

But I still left that conversation thinking there is a way to make a new math that would make more sense and be consistent aswell. 

 

 

Edited June 17 by Salvijus

[Ero]

6 minutes ago, zurew said:

I personally disagree with the idea that math is empirically grounded (meaning the truth value of math propositions are not directly depended on the physical laws of this universe) .

I think Leo is referring more to the fact that the origin of mathematics was grounded in everyday reality (i.e empiricism), which is kinda true. Multiplication and addition had a physical meaning before they were abstracted as binary operations, whereas with the current axioms you could theoretically define any arbitrary set with a multiplication table. For example, in the binary set Z/2, i.e the set of {0, 1}, you do indeed have 1 + 1 = 0. 

The same was true for Calculus when developed by Newton and Leibniz - they almost entirely used their physical intuition. It was developed into a rigorous mathematical discipline (Real Analysis) much later when inconsistencies, such as Weierestrass' s function, appeared as pathological.

That said, current mathematics operates in realms that are qualitatively different than reality. That doesn't make them less real or interesting, so you are generally right to question the idea that ALL of mathematics is empirical. The philosophical position that only mathematical objects that can be realized in reality is called finitism and has a host of problems on its own. Traditionally finitist reject the existence of infinity, imaginary numbers and a host of other very useful concepts which have allowed us to formulate Quantum Mechanics and General Relativity, which are clearly very successful (you having a screen to read this is further evidence). 

[Ero]

11 minutes ago, Salvijus said:

What makes no sense in math is that x^0=1. 

Although there was one mathematician here who explained how mathematicians are doing crazy mathematical gymnastics to make it so. Unfortunately I forgot his proof, haha. But it was pretty interesting.

But I still left that conversation thinking there is a way to make a new math that would make more sense and be consistent aswell. 

 

 

It follows from the fundamental properties of exponentiation, namely the additive property. When you have the same base, i.e a^b * a^ c, you can add the exponents : a^ b * a^ c = a^ (b+c). When you define negative powers as the reciprocals, i.e  a^ (-b) =  1/ a^ b, you can convince yourself that a^0 = 1:

a^ b * a^(-b) = a^(b-b) = a^0 = a^b* (1/a^b) = 1 due to cancellation. 

Edited June 17 by Ero

[zurew]

31 minutes ago, Ero said:

Multiplication and addition had a physical meaning before they were abstracted as binary operations, whereas with the current axioms you could theoretically define any arbitrary set with a multiplication table

Yeah thats probably true, that they maybe had some physical meaning, but my understanding is that  right now math is not depended on  any specific empirical fact. Like you could change all the empirical facts that we accept to be true right now, and math would still hold because of its axioms. In other words, I don't see any direct connection between any specific kind of empirical fact and between any specific kind of math axiom.

Because if the claim is that there is a direct dependency between a specific empirical fact and a specific math axiom, then my question would be ,which specific empirical fact change would necessitate the changing or the redefining of one of the math axioms?

Edited June 17 by zurew

[Salvijus]

6 minutes ago, Ero said:

It follows from the fundamental properties of exponentiation, namely the additive property. When you have the same base, i.e a^b * a^ c, you can add the exponents : a^ b * a^ c = a^ (b+c). When you define negative powers as the reciprocals, i.e  a^ (-b) =  1/ a^ b, you can convince yourself that a^0 = 1:

a^ b * a^(-b) = a^(b-b) = a^0 = a^b* (1/a^b) = 1 due to cancellation. 

This is a pretty cool demonstration but I remember I was proposing an alternative way of doing math. 

What I said to the other mathematician was. 

If 2^3 = 2*2*2 ( three 2's that multiple with each other) 

   2^2 = 2*2  (two 2's that multiple with each other) 

   2^1 = 2 (one 2) 

   2^0 = _____ (no answer. There are nothing that multiple with each other) 

   2^-1 = 2 (one 2 that is in division) 

   2^-2 = 1 (2/2) 

   2^-3 = 1/2 (2/2/2)

Etc. See what I mean? I thought it makes more sense like that. And then he somehow started bending things like an avatar lol. 

 

[Salvijus]

If 2^3 = 2*2*2 = 8

Then, 2^-3 = 2/2/2 = 1/2

A simpler summary of my math logic. 

But we are told in schools that 2^-3 = 1/8. That's when I quit school and never came back (:D joking) 

 

 

Edited June 17 by Salvijus

[Ero]

6 minutes ago, zurew said:

Yeah thats probably true, that they maybe had some physical meaning, but my understanding is that  math is not depended on  any specific empirical fact. Like you could change all the empirical facts that we accept to be true right now, and math would still hold because of its axioms. In other words, I don't see any direct connection between any specific kind of empirical fact and between any specific kind of math axiom.

Because if the claim is that there is a direct connection between a specific empirical fact and a specific math axiom, then my question would be ,which specific empirical fact change would necessitate the changing or redefining of one of the math axioms?

Ah, I see what you mean. Yes, you are correct, no disagreement here. The epistemological position of empiricism applies to physics, i.e providing experimental demonstration to a statement is sufficient to establish its truth. In mathematics, the epistemological approach is different, whereas you need to provide a "proof"  - a sequence of logical deductions based on collectively-established truths, namely axioms. These axioms can change and you can even construct something called "mathematical universes" where the same "proofs" lead to different statements. It gets really loopy.

To conclude with an example of why you are right, consider String Theory -  mathematically correct, but physically not even close (since we are not in 11 dimensions and the cosmological constant is positive not negative). Which is why people have won Fields medals and not Nobel Prizes, hah. 

[Ero]

@Salvijus So, what you are constructing is a different definition of exponentiation as a binary operation. When doing this, you always have to establish that is it is well-defined, i.e it is a homomorphism. Without drowning you in technicalities, consider that a binary operation is defined as a map , whereas when I give you two element, you return one based on the rule you have established (mapping two elements to one). In math, you can note this as a function, i.e 2^3 can be written as h(2, 3). Now, let us simplify by considering only when using base 2, as in all of your examples. Let this function be h2(x) = 2^x. Now, a necessary property when defining a binary operation, is to have that whether you perform the operation first within the parenthesis or after, it should be the same result (this is the meaning of a homomorphism - otherwise your operations is not well-defined), i.e h2(a + b) = h2(a)*h2(b). Using your current examples, let us consider a = 3 and b = -1. By what you wrote, h2(3) = 2^3 = 8 and h2(-1) = 2^-1 = 2. Then h2( 3- 1) = h2(2) = 2^2 = 4 but h2(3)*h2(-1) = 8*2 = 16, which is clearly not equal to 4. Hence, the operation you created is not well-defined. 

Edited June 17 by Ero

[Salvijus]

13 minutes ago, Ero said:

@Salvijus So, what you are constructing is a different definition of exponentiation as a binary operation. When doing this, you always have to establish that is it is well-defined, i.e it is a homomorphism. Without drowning you in technicalities, consider that a binary operation is defined as a map , whereas when I give you two element, you return one based on the rule you have established (mapping two elements to one). In math, you can note this as a function, i.e 2^3 can be written as h(2, 3). Now, let us simplify by considering only when using base 2, as in all of your examples. Let this function be h2(x) = 2^x. Now, a necessary property when defining a binary operation, is to have that whether you perform the operation first within the parenthesis or after, it should be the same result (this is the meaning of a homomorphism - otherwise your operations is not well-defined), i.e h2(a + b) = h2(a)*h2(b). Using your current examples, let us consider a = 3 and b = -1. By what you wrote, h2(3) = 2^3 = 8 and h2(-1) = 2^-1 = 2. Then h2( 3- 1) = h2(2) = 2^2 = 4 but h2(3)*h2(-1) = 8*2 = 16, which is clearly not equal to 4. Hence, the operation you created is not well-defined. 

Hmmm. That's a pretty cool demonstration also. 

I have nothing to say. Ggwp. 

I wonder if it's theoretically possible to take the model I presented and make it consistent aswell? So that it would be well defined throughout all math? 

[Ero]

1 hour ago, Salvijus said:

I wonder if it's theoretically possible to take the model I presented and make it consistent aswell? So that it would be well defined throughout all math? 

As long as positive exponents are defined as the repeated multiplication of the base (which is how it has been defined in the first place) the need for well-definedeness (homomorphism) necessitates that negative exponents are reciprocals. To see this just use the example from above, namely, let 2^(-1) = x be the unknown and let us solve for it: 
For a homomorphism, we need h2(a + b) = h2(a)*h2(b). Replacing h2(-1) = x, we get:
2^3 * 2^-1 = 8 * x = 2^(3-1) = 2^2 = 4, hence x = 4/8 = 1/2. 
 

[Salvijus]

1 hour ago, Ero said:

As long as positive exponents are defined as the repeated multiplication of the base (which is how it has been defined in the first place) the need for well-definedeness (homomorphism) necessitates that negative exponents are reciprocals. To see this just use the example from above, namely, let 2^(-1) = x be the unknown and let us solve for it: 
For a homomorphism, we need h2(a + b) = h2(a)*h2(b). Replacing h2(-1) = x, we get:
2^3 * 2^-1 = 8 * x = 2^(3-1) = 2^2 = 4, hence x = 4/8 = 1/2. 
 

This is solid proof.

So is the current model of mathematics the only model that can be consistent? There can’t be any alternative mathematics?

Because for some reason there is still a speck of hope there could be a way to make the alternative model work. All the mathematical rules would need to be adjusted tho. Like this rule wouldn't work the same way anymore in a new model I presented a^ b * a^ c = a^ (b+c). And you were using the normal version of this rule for your proof. That creates a problem. 

 

Edited June 17 by Salvijus

[Ero]

1 hour ago, Salvijus said:

Like this rule wouldn't work the same way anymore in a new model I presented a^ b * a^ c = a^ (b+c). And you were using the normal version of this rule for your proof. That creates a problem. 

 

What may this problem be? I would appreciate if you can formulate it similarly to how I showed you that without this “rule” you get inconsistent results from what should be the same operation (i.e it is not well-defined)

[Salvijus]

1 hour ago, Ero said:

What may this problem be? I would appreciate if you can formulate it similarly to how I showed you that without this “rule” you get inconsistent results from what should be the same operation (i.e it is not well-defined)

I don’t think I can do what you did. But I can try to clarify the problem I was pointing out in my own way. 

Suppose  2^3 * 2^-1 = 8 * 2 = 16

So this formula  a^ b * a^ c = a^ (b+c) is no longer true in a new model and it either has to be discarded or readjusted to fit a new model. 

The problem I was pointing at was you have successfully demonstrated that a new model is not well defined but you were using old formulas to do that. 

My question is. What if all the formulas and rules were to be changed and readjusted to fit the new model. And create entirely new definitions. Would that work? Is there a way to make new math so to speak? 

 

Edited June 17 by Salvijus

[gettoefl]

2^0 = 2^(1-1) = 2^1 * 2^-1 = 2 * 0.5 = 1

[Ero]

@Salvijus

The reason I asked what would the problem be was not to be a jackass about it, but because if you want to develop new math and be taken seriously, you want to be able to communicate what are the problems you are trying to fix and why we would want it.

You have to understand that math did not become what it is because someone said so or because the “annunaki” gave us multiplication tables or whatnot. Mathematics is a millenia-old endeavour (the oldest of all sciences) that has been rooted in all the basic abstractions we would want to have as humans, such as arithmetic, geometric objects and so on (most of which were formulated independently across cultures - Hellenic, Vedic, Arabic, Chinese, etc.). Many of the current fields, such as Complex Analysis, Algebraic Geometry, Representation Theory, etc. actually emerged from wanting to answer basic questions about arithmetic (such a highlight is Fermat’s Last Theorem - look it up, fascinating stuff). Furthermore, some of the smartest humans to have ever lived have spent their entire lives on it. Consider child prodigies that never stopped working on math, such as Terence Tao and Noam Elkies, who is one of my professors. The likelihood that you have found an inconsistency in arithmetic, crowned the “queen of mathematics” by Gauss is close to none. The “palace of mathematics” is one of the most solid “structures” in the collective intellectual pantheon. That said, that doesn’t mean there isn’t place for revolutionary work.

If you are familiar with Kuhn’s “The Structure of Scientific Revolutions”, you may recognize that math is not immune to paradigmic thinking, in fact the opposite - due to its size and complexity, majority of the subfields require specialization and an epistemic “trust” that the theorems proved in some adjacent field to yours are indeed correct. Novel work occurs at the interface between different fields, by demonstrating that different concepts are in fact one and the same (isomorphic) - Grothendieck’s Algebraic Geometry and the Langlands program are examples of such revolutionary work.

To conclude, due to the fundamentally social nature of mathematics, you have to understand that unless your model allows for all of current mathematics, whilst very clearly solving an inconsistency/paradox within it or very fundamentally reformulates a field, the likelihood of it being accepted as the new canon is non-existent. Whilst people like Terence Howard may be able to sway people who need to brush up on middle school math, he is never going to be accepted as revolutionary, because this type of “rewriting” of some of the most basic arithmetic concepts has already been evolutionarily discarded as part of the cycle of paradigm formation in mathematics as an epistemic endeavour.