> It truly breaks my heart to see any of the math snipped out yet again.

A false statement. I posted a link to the evidence, which cannot be summarized to satisfy your short attention span. Your unwillingness to read it only reveals your shallow grasp of modern technical topics.

Here is a short list of topics that cannot be converted into bumperstickers, your preferred medium of expression:

   Special and General Relativity

   Quantum Physics et. al.

   Cosmology

   Biology

   Modern Set Theory

This list is by no means complete, but the last entry overlaps with the topic of this conversation, for reasons that will not be obvious to you unless and until you overcome your distaste for evidence.

> I do wish you well and I'm sorry if my requirement for rigorous statements in math seems pointless to you.

Excuse, me, what? I have posted links to the evidence that proves your position to be false, but you won't read it. You are to modern times what a religious fundamentalist is to science -- the primary obstacle to human progress.

Am I saying your unwillingness to absorb ideas outside your short attention span is itself an obstacle to human progress? No, actually, I'm saying you should avoid drawing conclusions without first examining the evidence.

It isn't only that you won't read the evidence that proves your position to be wrong -- that is simply sad, not fatal. It is that you draw conclusions without first examining the readily available evidence.

> A set of theorems (of which I accept all you've named) without the definitions of exactly how and why they apply to the give problem is no more mathematics than a bunch of bricks is a chimney.

Wow. You just dismissed the validity of all the topics in my list above, each of which produce everyday valid conclusions derived from very complex logical and mathematical antecedents.

Get professional help.

> A set of theorems (of which I accept all you've named) without the definitions of exactly how and why they apply to the give problem is no more mathematics than a bunch of bricks is a chimney.

A second reply. What you're not getting is that the mathematical equations in Turing's paper don't make its point. There are plenty of equations, but the paper's meaning lies in its logical arguments, for which the equations can only play a supporting role.

If the equations are taken out of the paper and presented separately, as you have repeatedly demanded, the paper's thesis falls apart. But to understand this, you would have to read the paper itself and absorb Turing's logical arguments. And the paper can't be made shorter without losing its meaning.

More evidence for this is given by the fact that Alonzo Church published a similar paper in 1936, different author, different equations, but the same logical argument and conclusion, such that Church and Turing are now given equal credit for the basic idea -- an idea that is supported by equations but not provided by them (https://courses.fit.cvut.cz/MI-VYC/church-a-note-on-the-ents...).

This is true in many parts of mathematics and logic. Another example is Einstein's 1905 paper later identified as the source of "special relativity". If you remove the equations and present them separately, all meaning is lost. In fact, in that case, the actual meaning of a particular equation was lost to Einstein himself, but that meaning occurred to his former math teacher Hermann Minkoswski, who went on to publish about something Minkowski called "spacetime". As before, the equations didn't convey the paper's real meaning, they could only play a supporting role. About this outcome Einstein later said, "Since the mathematicians have invaded relativity theory, I don't understand it myself any more."

Another more recent example is the recently solved "Navier–Stokes existence and smoothness problem" as it was titled by the Clay Mathematics Institute. The equation is easy to render, but as with the other examples, the problem to be solved goes far beyond the equation itself, and the meaning of the recent result lies not in the equation, but in its treatment and processing by AI.

In Navier-Stokes, everyone had access to the equations, but until recently no one could answer a fundamental question about it, and as with the prior examples, the real meaning requires one to read the articles that provide the logical reasoning. In each of these examples and many more, listing the equations can only be a preliminary step to true understanding.

More importantly, the meaning of the papers cannot be summarized in fewer words than are provided by the papers themselves. Mathematicians don't go out of their way to make their papers longer than their contents require, in fact, quite the opposite.

I can't believe you're still expecting bumperstickers to stand in for technical articles. As before, you should read the original articles instead of complaining that they're too long for your limited attention span.

> There are plenty of equations, but the paper's meaning lies in its logical arguments

This is the difference I feared. I trust, understand, and take lead of the rigorous math of the paper, you opt to follow in English arguments of what you think it's supposed to mean or apply to. No matter how much math is discussed, it'll never lead to anything if that's not where your understanding of the paper comes from.

It's one of the main things which separate hard sciences from things like Psychology where anyone can just talk about what they understand/see or want to link instead of define it in logic itself, and I must admit I've never believed much of the soft sciences because of that lack of rigor. Maybe that rejection of soft science as truth is a limitation of mine :) but I don't see myself abandoning the view any time soon.

>> There are plenty of equations, but the paper's meaning lies in its logical arguments

> This is the difference I feared. I trust, understand, and take lead of the rigorous math of the paper, you opt to follow in English arguments of what you think it's supposed to mean or apply to.

Excuse me? The articles don't prove their theses with mathematics, they prove them with logic. Equations are assistants, but they're entirely replaceable, as proven by the fact that Church and Turing used different equations in support of their theses -- articles that come to the same conclusion.

How did you miss the significance of the Alonso Church article's final sentence: "The general case of the Entscheidungsproblem of the engere Funktionenkalkül is unsolvable." Where are the equations that you think make the point better than these words? Certainly not in the article. Why didn't Church refer to an equation to support his conclusion? The answer is that his conclusion is a logical one, not a mathematical one.

In the case of Navier–Stokes, the equation is a preliminary, a self-evident statement about energy, inertia, pressure and a few other things. If it were rewritten (as it often is), the problem remained to be solved. Those who solved it didn't post a new equation, they posted a new insight.

In the relativity example, Einstein wrote an equation but didn't understand it -- his math teacher took over. Any number of equivalent expressions would have provided a basis for progress toward a more comprehensive theory. The point was the ideas, not the equations.

> ... the rigorous math of the paper ...

Nonsense. In both papers, the authors use mathematics only to support their points, in the same way that an author uses words to craft a story. If separated from the logical thread, the words (the equations) lose all meaning. This is proven by the fact that the two papers use different mathematics to support the same thesis.

But I see I'm wasting my time. Mathematics is a language, but to use it, you must have something to say.

Church and Turing did not prove their papers with the words, they described the math proving their statements in them and what that math implies (which is why the ending summary would not be math, it must already have been shown to be a summary). In however many comments you have certainly used a lot of words to describe your thought but have given nothing in the way of the math used to base the difference in your claim between that of Church/Turings.

A psychologist could just as well assert the halting problem contains math applicable to any system and therefore blue cars mathematically make us angry. The doubt would then not be on Turing but on the psychologist to show how on Earth they are mathematically linking what they are saying with what Turing actually showed. That they don't see a requirement in having the chain of math rigorously define and prove the claim when they make it is precisely why they are a soft science - they can just claim it seems to relate by less rigorous means. Defend that kind of approach if you wish, it does not make it more rigorous.

I expect we'll soon hear words about more mathematicians and scientist, related to the current problem or not, rather than math. Perhaps I've allowed myself baited by hope again on the mention of what math is needed being discussed.

> Church and Turing did not prove their papers with the words,

THAT STATEMENT IS FALSE. The two articles used DIFFERENT EQUATIONS TO SUPPORT THE SAME LOGICAL ARGUMENT. Church and Turing are jointly credited with the same logical conclusion, even though they used different mathematical arguments. NO ONE said, "Wait, the equations are different, therefore the conclusion must be different." At least no one with an education.

> ... they described the math proving their statements in them and what that math implies (which is why the ending summary would not be math, it must already have been shown to be a summary).

Church and Turing made the same logical claim, using the same words but different mathematical arguments. This places the math in a supporting role, below logical reasoning. This is true in all the examples I provided. Math is a language, but to use it, you must have something to say.

In the Relativity example I gave, Einstein's math teacher used the SAME equation to draw a DIFFERENT CONCLUSION, because EQUATIONS SUPPORT IDEAS, THEY AREN'T THEMSELVES IDEAS.

Just as with words, math equations never stand by themselves -- they support ideas, and the same idea can be expressed by different mathematical arguments. That was the point of all my examples, which clearly fell on the ears of a religious true believer.

This equation -- y = sqrt(1-x*x) -- can be taken to mean (a) pi/4 if integrated on the interval 0 < x < 1, or (b) space and time are separate dimensions, which is what Minkowski noticed and Einstein didn't. But by itself, THE EQUATION MEANS PRECISELY NOTHING -- for that, you must have ideas.

If you possessed the intellectual integrity of the average 12-year-old, you would realize that using the same words as Ernest Hemingway doesn't make you a great writer, and using the same equations as Einstein doesn't make you an original thinker. That requires ideas, not equations.

I post this for the benefit of other readers, who unlike you might have some depth of understanding, some grasp of the relation between ideas and equations.