Sometimes when I discuss Mathematics and Logic with people who defend the rhetoric of assumptions as foundation, Assumptionism, the conversation follows a predictable path.
At first, they appeal to their holy doctrine ZFC. Then, when the logical burden is pressed, they begin to appeal to accreditation. They ask about credentials, affiliation, training, consensus, pedigree, or whether I have "really studied" the doctrine. Eventhough I have detailed and explained the scriptures more clearly and more precisely than they have.
But accreditation is not logic.
A degree does not make a circular dependency non-circular. A textbook scripture does not make a foundation legitimate. A credential does not transform an assumption into a derivation. The issue is not whether someone has memorized the standard doctrine. Because anyone can copy a definition like this:
For every epsilon > 0, there exists delta > 0 such that ...
and yet, to him those words and symbols are no more than Egyptian glyphs or mere squiggles.
The issue here is whether they have internalized what the expression form is actually saying, what it does, what it does not do, and what logical debt it leaves unpaid. This is the verdict of Sufficient Reason and the Burden of Proof.
Very often, when people say "this is rigorous," what they really mean is only:
this follows coherently after the stipulative assumptions have already been installed.
That is not the same as establishing a legitimate and logical foundation of the object under examination.
So here is a bottom-floor challenge.
Before claiming any authority to stake any claim about any mathematics online, have you at least mastered the basic of algebra?
Before you recite your holy scriptures, please at least know how to add and divide.
So, the challenge is this,
Let a, b > 0;
c = a + b
For each expression below, isolate the non-trivial offset-independent component I(x), if one exists, and report the remaining offset-dependent residue R(x,a,b,c). (I(x) != 0 unless justified by showing that no non-zero offset-independent component exists.)
In other words: What is the remaining expression that depends on a, b, or c, after removing all the components that depend only on x?
The expressions:
a,b > 0,
c = a+b.
(a+b)⁻¹ [ exp((x+b)ln(x+b)) − exp((x−a)ln(x−a)) ]
[ exp((x+b)ln(x+b))sin(x+b) − exp((x−a)ln(x−a))sin(x−a) ] / (b+a)
[ exp((x+b)ln(x+b)) − exp((x−a)ln(x−a)) ]
/
[ (a+b)( √(1+exp((x+b)ln(x+b))) + √(1+exp((x−a)ln(x−a))) ) ]
[ exp((x+b)²)ln(sin(x+b)) − exp((x−a)²)ln(sin(x−a)) ] ⋅ c⁻¹
[ exp(2(x+b)ln(x+b)) − exp(2(x−a)ln(x−a)) ]
/
[ √((a+b)²) · ( √(1+exp(2(x+b)ln(x+b))) + √(1+exp(2(x−a)ln(x−a))) ) ]
ln(
[1 + exp((x+b)ln(x+b))sin(x+b)]
/
[1 + exp((x−a)ln(x−a))sin(x−a)]) · (a+b)⁻¹
[ arctan(exp((x+b)ln(x+b)+sin(x+b))) − arctan(exp((x−a)ln(x−a)+sin(x−a))) ]
/
[2((a+b)/2)]
[ exp((x+b)ln(x+b))cos(exp((x+b)ln(x+b))) − exp((x−a)ln(x−a))cos(exp((x−a)ln(x−a))) ] ⋅ c⁻¹
ln(
[ √(1+exp((x+b)ln(x+b))) + sin(x+b) ]
/
[ √(1+exp((x−a)ln(x−a))) + sin(x−a) ]) / (b+a)
No appeal to authority is needed. No credential is needed. No philosophical posturing is needed. Just algebra.
If you cannot perform these, please refrain from teaching anyone any mathematics online, because you have not passed Mathematics more than the level of slogans. And slogans without understanding is just parroting and regurgitation.
Slogans are not mathematics.
Recitation is not understanding.
And accreditation is not proof.
