Why does 2+2 = 2×2?

[nyquistfrequency]

We all know the elementary algebraic explanation for this identity. If we set addition equal to multiplication:

$$ x+x = x \cdot x $$
$$ 2x = x^2 $$
$$ x^2 - 2x = 0 $$
$$ x(x - 2) = 0 $$

Discounting zero, the only possible value that satisfies this condition is $ x = 2 $.
But this explanation only describes HOW $ 2+2 = 2 \cdot 2 $ but not WHY.
Because we have been treating the number 2 as a purely discontinuous numerical quantity, we have completely missed the profound qualitative reality behind the mechanics of number.
From the purely quantitative perspective this equation comes to be seen as a coincidence, an strange quirk with no deeper meaning behind it.

In Pythagorean mathematics, numbers are expressions of qualitative principles, not just quantities for counting. The behaviors of numbers are manifest expressions of unmanifest cosmic principles. Just as a decimal expansion is only "correct" in so far as it adheres to an simultaneous irrational magnitude like $\sqrt{2}$, the manifest world finds its order solely through an unmanifest Principle. These decimal representations being without end necessitates their incompleteness, so therefore, while the $\sqrt{2}$ must be completely absent from the decimals, it is the principle of their necessity, and this intrinsic necessity that something tries to fulfill is known as its purpose. From the worldly perspective of one who tries to reduce everything to discontinuous quantity, the $\sqrt{2}$ is "Non-existent" (because it isn't a numeral but an incommensurable magnitude), yet, it is the most "Real" thing because it governs the truth of every one of its decimal approximations.

Now Addition is an expression of succession (the mechanical "how" the means) a step by step accumulation happening linearly through time ($x+x+x+...$) Multiplication, on the other hand, represents simultaneity (the archetypal "why" the ends), the higher principle which governs the entire addition operation simultaneously ($ x \cdot C = x+x+x+... $).

Now we may notice here that the succession of additions $x+x+x+...$  can never reach the principle of multiplication which governs all those additions simultaneously so one can never reach $ x \cdot x $ by adding $ x + x $ to itself. And this is the secret of incommensurability, that succession cannot reach the simultaneity form which governs the successive motion, the "how" can never reach the "why".

But notice in $ 2\cdot2 = 2+2 $ number 2, the Dyad, the very first qualitative distinction, is the only absolute structural exception in Reality. The Dyad represents the very distinction between means and ends, how and why, substance and essence and so is the only level where succession and simultaneity are unified.

[StudentDriver]

Stupid question deserves Stupid answer: 2+2=2*2 bc both equals 4.

[nyquistfrequency]

Stupid question deserves Stupid answer: 2+2=2*2 bc both equals 4.

Yay, you have successively discovered WHAT they equal but NOT WHY they are equal.
You have grasped THAT they arrive at the same destination but NOT WHY they do and what this says about the deeper nature of the number 2.
It's quite obvious you didn't even read my post or even understand the title of my post.

[StudentDriver]

Aight, How tf does addition equal multiplication. Because 100*5=500 and 100+5=105. There is a literal big difference between addition and multiplication.

[nyquistfrequency]

Aight, How tf does addition equal multiplication. Because 100*5=500 and 100+5=105. There is a literal big difference between addition and multiplication.

It's quite clear that you did not properly read or understand what I wrote.
I did not say addition and multiplication are the same, I was saying that multiplication is addition held in simultaneity. for x+x+x+... = x + C where C is some arbitrary constant.
What you would have to do successively with addition you can do simultaneously with multiplication. And succession can never catch up to simultaneity, so the only point where succession and simultaneity are one, is 2 for 2+2 = 2×2. So 2 the Dyad is the exact point of division into succession and simultaneity, substance and essence.

By pointing out that 100×5 doesn't equal 100+5, you are simply proving that once you move past the number 2, the split between succession and simultaneity becomes massive.

[StudentDriver]

Ah my bad, so you are setting x+x= x*x and finding solutions to this equation? So what's so special about 2? Explain to me like I'm 5.

[MagiOmni]

I dont think I agree with the premise that using algebra or computational reduction to solve for variables is fundamentally different or even uninteresting compared to some abstract notion of the "essence of an expression", and as a matter of fact I think algebra takes the notion of the "essence of an expression" and makes it more rigourous. You take for granted certain properties of algebraic expressions and ignore the beauty behind the actual algebra you're doing. Going from the first step to the second step you substitute addition for multiplication and multiplication for exponentiation, a property that comes from how those operations definitionally reduce as well as how equality works (two expressions are equal if the truth of any proposition about one implies the truth of the same proposition about the other). From the second to the third step, you use the property of equality where if x = y then f(x) = f(y), true using substituting x for y in f(x)=f(x), which is only true due to functions mapping a single value to another single value, as well as taking advantge of a symmetry of addition, adding and subtracting the same number is the additive identity (0). From step 3 to 4, you take advantage of the commutativity of multiplication and once again the substitution property of equality. To get the solutions from that last equation, you take advantage of the fact that multiplication by zero is always zero as well as the property that linear equations always have a single zero. We treat algebra as a methodology so much we ignore its rigourous roots in symmetries and the fundamental properties of math and take forgranted the beauty that underlies it all.

[xleph]

How does your mental model of essence and substance translate to tetration or higher compositions ("simultaneities") of addition, multiplication, et cetera?

Also your LaTeX as above, if I have formatted properly, is:

[nyquistfrequency]

How does your mental model of essence and substance translate to tetration or higher compositions ("simultaneities") of addition, multiplication, et cetera?

Also your LaTeX as above, if I have formatted properly, is:

Every degree of simultaneity works in the same way, no matter how far you go.
Simultaneity and succession are relative to one another, there is always a greater simultaneity, but no perfect simultaneity.

It is only logical that if we were to play the totality of possible sounds simultaneously, there would always be one sound which would be the exact opposite polarity of another, causing each and every sound to cancel with its polar opposite resulting in silence. So the totality of sound is not any particular sound.

But is all sound in totality devoid of all form and consequently manifest existence?
That would clearly contradict our experience of sound itself. If absolute silence were merely the sum of all waves, we would have destroyed the possibility of sound entirely, yet we hear sound, and silence is the background of every sound. All we have realized here is that all possibility cannot be evaluated to an undifferentiated single unified manifestation. Therefore the totality of Being has no particular existence in of itself, the "ability to be" although a reality, is not any particular being.

- Addition: $2 + 2 = 4$
- Multiplication: $2 \times 2 = 4$
- Exponentiation: $2^2 = 2 \uparrow 2 = 4$
- Tetration: $^2 2 = 2 \uparrow\uparrow 2 = 4$
- Pentation: $2 \uparrow\uparrow\uparrow 2 = 4$
- Hexation: $2 \uparrow\uparrow\uparrow\uparrow 2 = 4$
- All Simultaneity: $2 \uparrow^n 2 = 4 \quad \text{(for any integer } n \geq 1\text{)}$

We can formally prove $2 \uparrow^n 2 = 4$ for all $n \geq 1$ using the formal recursive definition of Knuth's up-arrow notation:

$$a \uparrow^n b = a \uparrow^{n-1} (a \uparrow^n (b-1))$$

With the baseline rule that $a \uparrow^n 1 = a$.

By definition, a single up-arrow is standard exponentiation:

$$2 \uparrow^1 2 = 2^2 = 4$$

The base case holds.

Assume the statement is true for some integer $k \geq 1$. That is, our inductive hypothesis is:

$$2 \uparrow^k 2 = 4$$

Now, we must prove it holds true for $k + 1$, meaning we want to show $2 \uparrow^{k+1} 2 = 4$. Let's expand this using the recursive rule:

$$2 \uparrow^{k+1} 2 = 2 \uparrow^k (2 \uparrow^{k+1} 1)$$

Using the baseline rule $a \uparrow^n 1 = a$, we know that $2 \uparrow^{k+1} 1 = 2$. Substituting that back into our equation gives:

$$2 \uparrow^{k+1} 2 = 2 \uparrow^k 2$$

By our inductive hypothesis, we already know that $2 \uparrow^k 2 = 4$. Therefore:

$$2 \uparrow^{k+1} 2 = 4$$

By mathematical induction, the identity $2 \uparrow^n 2 = 4$ is proven true for any integer $n \geq 1$.

[nyquistfrequency]

I dont think I agree with the premise that using algebra or computational reduction to solve for variables is fundamentally different or even uninteresting compared to some abstract notion of the "essence of an expression", and as a matter of fact I think algebra takes the notion of the "essence of an expression" and makes it more rigourous. You take for granted certain properties of algebraic expressions and ignore the beauty behind the actual algebra you're doing. Going from the first step to the second step you substitute addition for multiplication and multiplication for exponentiation, a property that comes from how those operations definitionally reduce as well as how equality works (two expressions are equal if the truth of any proposition about one implies the truth of the same proposition about the other). From the second to the third step, you use the property of equality where if x = y then f(x) = f(y), true using substituting x for y in f(x)=f(x), which is only true due to functions mapping a single value to another single value, as well as taking advantge of a symmetry of addition, adding and subtracting the same number is the additive identity (0). From step 3 to 4, you take advantage of the commutativity of multiplication and once again the substitution property of equality. To get the solutions from that last equation, you take advantage of the fact that multiplication by zero is always zero as well as the property that linear equations always have a single zero. We treat algebra as a methodology so much we ignore its rigourous roots in symmetries and the fundamental properties of math and take forgranted the beauty that underlies it all.

You have no clue as to the reason for WHY these fundamental properties you listed are valid, yet you list off all these "fundamental properties of math": a sequence of mechanics - identities, commutativity, substitution - as if listing the rules of a game explains WHY the game is allowed to exist in the first place. However, if you don't know WHY you end up treating math as an arbitrary rule book for methodology (HOW), and no true understanding can be gained in this way, for one can never reach the simultaneous WHY through the HOW of succession.

Richard Courant (a student of Hilbert who later founded an influential institute bearing his name) and H. Robbins wrote in 1941 in their introduction to What Is Mathematics? "A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from definitions and postulates that may be created by the free will of the mathematician." If that were true, they say, "mathematics could not attract any intelligent person. It would be a game with definitions, rules, and syllogisms, without motive or goal." In short, viewing math as an arbitrary formal game "without motive or goal" makes it meaningless, and once meaning is lost so is intelligibility and consequently Intelligence.

And this is why I stand with Ramanujan when he asserted, "An equation means nothing to me unless it expresses a thought of God." This is the exact same position as the Platonists and Pythagoreans. This is clearly evident in Proclus' statement about Euclid that "the whole of the geometer's argument is concerned with the cosmic figures", and by "cosmic figures", he means figures that belong to the order of the kosmos, that every proposition of geometry participates in cosmic order insofar as it expresses intelligible form. They saw mathematics as symbolic of a higher order, the laws of the permanent essence behind the impermanent appearances within existence, and they achieved lots of real groundbreaking mathematical work. It is in this way that mathematics is Symbolism and Symbolism is the Calculus of Meanings, a Symbol represents the limit of where some indefinite becoming approaches its telos or finite reason for being. Anyways I could care less what the consensus of mathematicians say because it is not them who determine what is True.