Why does 2+2 = 2×2?

[StudentDriver]

You seem to see more significance in the number 2 than the word "I". I put more emphasis on the word "I". Quite literally, know thyself. I am pretty sure, knowing thyself is more significant than knowing more about 2. And to Explain why the rules of a game is allowed to exist in the first place, look at what u r doing, literally... and philosophize around that.

And About the arbitrary rule book for methodology (HOW).... you do realize you can do that right? You can make sht up? (we can have arbitrary mappings between symbols). However we are interested in structures/algebra that can be described in a handful amount of statements (axioms). But there can be "algebra" (I mean I define algebra this way)... where an operator cannot be fully described by some general statement (axiom) ... "The computation with the operator is a case-by-case operator " <--- Doing things case-by-case. Imagine you have infinite memory... would you need axioms to do algebra? No. Hopefully that makes sense.

-And broski... you are possessed. You are connecting ideas where it is not meant to be. It is much simpler. Radically simpler.

[MagiOmni]

You have no clue as to the reason for WHY these fundamental properties you listed are valid

Yes I do? All the properties I listed are either basically definitionally true (like substitution) or derivable from their definitions (commutativity, distributivity etc.). I don't view mathematics as simply a set of rules, the point of my comment was that underlying the presumably elementary algebra were those fundamental truths, a clear why to a how. You need to clarify what exactly you mean by why vs how, or else I can't argue or agree with a point I apparently don't understand. What follows is a series of guesses at your true point and refutations of those guesses: If your point is that god is responsible for logic making sense, then sure? But if that's what you meant then your assertion that no true understanding can be reached without acknowledging gods responsibility in creating logic is false. Understanding about god is severely lacking in the modern mathematical community and yet we're continuing to see mathematical breakthroughs; you can view logic as not descending from anything else and gain an understanding of logic, the same way you can view god as not descending from anything else and gain an understanding of god. If your assertions are about why the axioms we assume are true, then again you can have an understanding of the axioms and the things that follow from them without understanding where they originate from, though most axioms are intuitively understandable. You can argue that the understanding we can achieve is limited by our understanding of the origin, sure, but when it comes to the foundations of math we already have, knowledge about them isnt inhibited by a (lack of) knowledge of god, except to the extent by which god grants humans the ability to acquire knowledge.

[nyquistfrequency]

You have no clue as to the reason for WHY these fundamental properties you listed are valid

Yes I do? All the properties I listed are either basically definitionally true (like substitution) or derivable from their definitions (commutativity, distributivity etc.). I don't view mathematics as simply a set of rules, the point of my comment was that underlying the presumably elementary algebra were those fundamental truths, a clear why to a how. You need to clarify what exactly you mean by why vs how, or else I can't argue or agree with a point I apparently don't understand. What follows is a series of guesses at your true point and refutations of those guesses: If your point is that god is responsible for logic making sense, then sure? But if that's what you meant then your assertion that no true understanding can be reached without acknowledging gods responsibility in creating logic is false. Understanding about god is severely lacking in the modern mathematical community and yet we're continuing to see mathematical breakthroughs; you can view logic as not descending from anything else and gain an understanding of logic, the same way you can view god as not descending from anything else and gain an understanding of god. If your assertions are about why the axioms we assume are true, then again you can have an understanding of the axioms and the things that follow from them without understanding where they originate from, though most axioms are intuitively understandable. You can argue that the understanding we can achieve is limited by our understanding of the origin, sure, but when it comes to the foundations of math we already have, knowledge about them isnt inhibited by a (lack of) knowledge of god, except to the extent by which god grants humans the ability to acquire knowledge.

If our fundamental properties are true purely because we made some definitions and assume those definitions are true, then there is nothing to distinguish them from an arbitrary dogma. To say that an unfounded assumption is "true" simply because it is internally consistent is to say that ultimately nothing is true - because all dogmas would be equally true, even mutually incompatible ones. This is the inevitable result of a secular philosophy that denies God, the Absolute, who is the singular source of all that is True.

This misunderstanding of truth, logic, and mathematics is what allows certain modern academics to rightfully say "math is racist". For if mathematics and logic is nothing more than a localized game played with arbitrary, subjective axioms that human beings invented for utility or persuasion, then it is entirely a cultural artifact. And if it is a cultural artifact, then it is inherently bound to the power dynamics, biases, and historical sins of the culture that codified it. Under this definition and misunderstanding, the academics who claim "math is racist" are not making an error, because logical and mathematical proofs are no longer seen as invariant geometric properties of reality, they are now treated as an arbitrary language framework imposed by a dominant group to maintain institutional power.

Furthermore, if a person argues that truth is just a system built on arbitrary assumptions, that we can simply pick different axioms to have different truths, and that the Münchhausen Trilemma proves no absolute proof exists - they are asserting an absolute proof. Because that proof is itself yet another proposition, it must by its own rule fall into infinite regress, circularity, or arbitrariness. The claim that "all proof is relative" is completely dismantled by the fact that the claim itself would require an absolute proof to be binding. This proves out of sheer logical necessity that there are foundational truths which do indeed prove themselves. For if the Trilemma is absolutely true, then absolute truth exists, and the Trilemma is false. If the Trilemma is only "relatively" true, then it doesn’t actually prove anything - it is just another arbitrary opinion with no binding power over necessary truths.

You claim you don't understand my definition of Why and How, but the reality is that I already defined it explicitly in my original post. I fear that in explaining this again to you, I might be trying to explain color to a person blind from birth, for a simultaneous quality can never be derived from a relational process. You cannot build a timeless essence out of sequential steps, no matter how many steps you take. If you cannot see past the "How" of the mechanical process to the "Why" that guides it, then the distinction will always appear non-existent to you as you conflate the two.

Let me state it again, How is succession whereas Why is simultaneity, I can even explain this in terms of calculus. How is the result of differentiation, it is how the hand must move in succession vector by vector, to draw the curve, whereas why is when one integrates that motion to realize what is being drawn, which was why the hand was moving. In regular life this distinction manifests as the means you take towards some ends. The how is the expression and the why is the meaning it is attempting to express.

Think of written language: a single pen stroke is meaningless. Letters gain sense only when combined into words; words only when arranged in a sentence; a sentence only when it participates in the coherence of a paragraph. Ultimately, the text only means anything to the degree that it approximates the unmanifest meaning it was attempting to express, just like $1.414...$ approximates the incommensurable magnitude of $\sqrt{2}$.

If you need more examples, just read my original post, and there's one I wrote on Cantor's Diagonalization I would advise checking out as well.

When you treat math as an arbitrary rulebook, your "progress" is built on quicksand. If your foundational axioms contain a hidden error, it is only a matter of time before the consequences rear their ugly head and render your entire life's work meaningless - much like when Bertrand Russell dropped his famous paradox on Gottlob Frege, instantly shattering Frege's entire life's work on the foundations of arithmetic.

On the other hand, when an intellect approaches the timeless Why of Simultaneity, it can immediately see the shortest most elegant path to mathematical expression. But the pure formalist, who cannot see the why through the how, simultaneity through succession, is forced to drag themselves through an unnecessary tedious labyrinth of mechanical, sequential steps, completely oblivious to a much simpler proof. When one cannot see beyond the process, one might get good at predicting how the hand moves, but will never know what is being drawn - which is the entire reason why the hand was moving in the first place.

[MagiOmni]

If our fundamental properties are true purely because we made some definitions and assume those definitions are true, then there is nothing to distinguish them from an arbitrary dogma.

The way I view the entirety of math is the same way I view individual sections of abstract algebra. Those sections only hold true when they correspond to another truth. In the same way the assumptions of groups don't hold for sets and all functions between them and so theorems relating to groups don't apply to sets and all functions between them, if you create incorrect axioms for a foundation of mathematics that don't apply to reality then your conclusions won't apply to reality. While when it comes to abstract algebra it's easier to determine whether something fits a description because you're already working under the assumptions of a mathematical framework, when it comes to determine whether a mathematical framework fits a description of reality, that's not something that can be mathematically determined as it is quite literally outside the domain of mathematics. While it isn't true to say that truth is built on arbitrary assumptions, it is true to say that foundations of math, purely as a conceptual idea, are built on (potentially) arbitrary assumptions. Useful math that actually applies to reality, on the other hand, must be built on fundamental truths that, again, are outside the domain of mathematics, but rather in the domain of god.

You claim you don't understand my definition of Why and How, but the reality is that I already defined it explicitly in my original post. I fear that in explaining this again to you, I might be trying to explain color to a person blind from birth, for a simultaneous quality can never be derived from a relational process. You cannot build a timeless essence out of sequential steps, no matter how many steps you take. If you cannot see past the "How" of the mechanical process to the "Why" that guides it, then the distinction will always appear non-existent to you as you conflate the two.

Let me state it again, How is succession whereas Why is simultaneity, I can even explain this in terms of calculus. How is the result of differentiation, it is how the hand must move in succession vector by vector, to draw the curve, whereas why is when one integrates that motion to realize what is being drawn, which was why the hand was moving. In regular life this distinction manifests as the means you take towards some ends. The how is the expression and the why is the meaning it is attempting to express.

If I understand you correctly (which if my previous point makes sense then I believe I do), then I disagree with your characterizations of your definition whys and hows. The whys and hows aren't characterized by any operation or relations, but in if the math and logic that've been built up as a whole map to reality, while the how is math itself. Of course, I also believe using terms like why, how, simultaneity and succession are misleading when trying to get your point across as (again, if I understand correctly) words like why or how don't relate well to the point you're trying to get across, simultaneity doesn't mean anything when it comes to reasoning (formal reasoning is linear) and I still don't even understand what you were trying to mean by succession. Also, it'd be of advantage to you to not engage in antisocial behavior such as insults when someone is trying to understand the ideas you're trying to communicate.

On the other hand, when an intellect approaches the timeless Why of Simultaneity, it can immediately see the shortest most elegant path to mathematical expression. But the pure formalist, who cannot see the why through the how, simultaneity through succession, is forced to drag themselves through an unnecessary tedious labyrinth of mechanical, sequential steps, completely oblivious to a much simpler proof. When one cannot see beyond the process, one might get good at predicting how the hand moves, but will never know what is being drawn - which is the entire reason why the hand was moving in the first place.

(again, if I understand your point correctly) I don't disagree with this entirely, however I don't think it's the whole story. I think the full idea is explained best by Mr. Cromwell in this discussion, which you happened to have replied to so I assume you've read it before.
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[StudentDriver]

To nyqueistfrequency, you say a single pen stroke is meaningless, but to me is super profound. Thanks to freewill. (not the unrestrained freewill but there is definitely agency. If you are not aware of this, then u r a bot xD)

Btw I think you didnt really understand my "Why is 1+1=2 true" post. You needed a crazy sht ton of justification just to explain why 2+2=2*2 lmao.

[MathSchizo]

I do not fully understand your point or what you are trying to say but maybe you will know why 9+9=18 and 9x9=18 but in reverse? correct me if im wrong but x is just + repeating so if you do 2 + 2 + 2 = 2x3 since there is 3 2s?