The Limits of Understanding

[Native]

Native said: ↑
Oh, so "logics of math and philosophy" don´t affect anything in science?

You STILL have to explain WHY scientists are intellectually/scientifically shocked by the Godel "logics of math and philosophy".

Why is that?

Ask them.

Well, I otherwise got the impression that your mathematical skills could explain the "Godel logics of math and philosophy" and it´s "shocking implications" for the standing cosmological science.

 

[Polymath257]

Native said: ↑
Oh, so "logics of math and philosophy" don´t affect anything in science?

You STILL have to explain WHY scientists are intellectually/scientifically shocked by the Godel "logics of math and philosophy".

Why is that?

Well, I otherwise got the impression that your mathematical skills could explain the "Godel logics of math and philosophy" and it´s "shocking implications" for the standing cosmological science.

I consider Godel's results to be shocking for philosophy and for certain areas of math. I don't consider them relevant to cosmological science at all. From what I can see, those who think Godel's results say something about cosmology either don't understand Godel's results or they don't understand cosmology, or both.

 

[Native]

Yes, the great advantage of math is that it is a *formal* language which attempts to limit the ambiguities of natural languages. But it is still a human language, made to help us explain the world, but branching out into poetry in modern mathematics.

So now mathematical number acrobatics are "poetry"?

For your information - Poetry - Wikipedia -:
"The oldest surviving epic poem, the Epic of Gilgamesh, dates from the 3rd millennium BCE in Sumer (in Mesopotamia, now Iraq), and was written in cuneiform script on clay tablets and, later, on papyrus. A tablet #2461 dating to c. 2000 BCE describes an annual rite in which the king symbolically married and mated with the goddess Inanna to ensure fertility and prosperity; some have labelled it the world's oldest love poem. An example of Egyptian epic poetry is The Story of Sinuhe (c. 1800 BCE).

Other ancient epic poetry includes the Greek epics, the Iliad and the Odyssey; the Avestan books, the Gathic Avesta and the Yasna; the Roman Virgil's Aeneid (written between 29 and 19 BCE); and the Indian epics, the Ramayana and the Mahabharata. Epic poetry, including the Odyssey, the Gathas, and the Indian Vedas, appears to have been composed in poetic form as an aid to memorization and oral transmission in ancient societies".
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Now, here you have real philosophical poetry described in and with written human languages And if you take these to have "ambiguities", it must derive from your lack of understanding its natural contents.

 

[Polymath257]

So now mathematical number acrobatics are "poetry"?

If you think that math is simply number acrobatics, you only show you don't understand much mathematics.

Yes, I use the word 'poetry' to describe the underlying beauty and structure of mathematics itself.

 

[Native]

I consider Godel's results to be shocking for philosophy and for certain areas of math. I don't consider them relevant to cosmological science at all. From what I can see, those who think Godel's results say something about cosmology either don't understand Godel's results or they don't understand cosmology, or both.

Or maybe you just don´t wish to deal with the annoying implications - or because you can´t see these to have any affects in the modern cosmological models.

It never can be "shocking for philosophy" as this just deals with natural facts in everything. On the other hand, it could be very shocking for math when this cannot describe natural facts philosophically everywhere.

 

[Native]

Yes, I use the word 'poetry' to describe the underlying beauty and structure of mathematics itself.

Oh well. It´s just a human made number construct anyway

 

[Polymath257]

Or maybe you just don´t wish to deal with the annoying implications - or because you can´t see these to have any affects in the modern cosmological models.

Modern cosmological models are described by differential equations (so, by the way, are the equations of E&M). The solutions to these differential equations are not impacted by the types of problems Godel was investigating.

It never can be "shocking for philosophy" as this just deals with natural facts in everything. On the other hand, it could be very shocking for math when this cannot describe natural facts philosophically everywhere.

It was shocking for philosophy because of the assumptions of philosophers like the logical positivists.

 

[Native]

It was shocking for philosophy because of the assumptions of philosophers like the logical positivists.

I don´t know if the 4 participants in the OP Video are "logical positivists philosophers" but the mathematical educated particians clearly expressed their "Godel"-shock.

 

[Polymath257]

I don´t know if the 4 participants in the OP Video are "logical positivists philosophers" but the mathematical educated particians clearly expressed their "Godel"-shock.

They certainly discussed the logical positivists. And they certainly did NOT say why they thought that Godel's results were relevant to physics. They did for math and philosophy, by the way.

 

[Native]

They certainly discussed the logical positivists. And they certainly did NOT say why they thought that Godel's results were relevant to physics. They did for math and philosophy, by the way.

Well, that why I asked you in the first place and so far, you haven´t convinced me why the Godel´s results are shoking.

 

[Polymath257]

Well, that why I asked you in the first place and so far, you haven´t convinced me why the Godel´s results are shoking.

At the time Godel worked, there was a program at the foundation on mathematics to prove it to be consistent (in other words, that it never produces a contradiction). There was another goal in this program to find a collection of axioms that was complete: it could answer any question in math (although finding the answer would still be difficult).

So, for example, it was proved that propositional logic is both consistent and complete. The same is true of predicate logic. Other extensions, like the axioms for real complete fields were shown to be complete.

This program was instituted by Hilbert, one of the best mathematicians at the time and there was a significant amount of work done towards this goal.

Godel showed that BOTH goals: completeness and a proof of consistency, are impossible.

THAT's what was shocking to those at the time. it struck at some very deep questions in what sorts of things are provable in math, which had impact on the philosophy of math.

 

[paarsurrey]

At the time Godel worked, there was a program at the foundation on mathematics to prove it to be consistent (in other words, that it never produces a contradiction). There was another goal in this program to find a collection of axioms that was complete: it could answer any question in math (although finding the answer would still be difficult).

So, for example, it was proved that propositional logic is both consistent and complete. The same is true of predicate logic. Other extensions, like the axioms for real complete fields were shown to be complete.

This program was instituted by Hilbert, one of the best mathematicians at the time and there was a significant amount of work done towards this goal.

Godel showed that BOTH goals: completeness and a proof of consistency, are impossible.

THAT's what was shocking to those at the time. it struck at some very deep questions in what sorts of things are provable in math, which had impact on the philosophy of math.

" what sorts of things are provable in math "

Can one apprise us, the ordinary persons, what are those things that are provable in math and those that not provable, please?

Regards

 

[Polymath257]

" what sorts of things are provable in math "

Can one apprise us, the ordinary persons, what are those things that are provable in math and those that not provable, please?

Regards

Well, mathematics has a set of basic assumptions, which we call the axioms. The modern set of axioms deals with the properties of sets. All of the rest of math is defined in terms of sets, including numbers, functions, vectors, geometry, etc. EVERYTHING is ultimately a set in modern mathematics.

So, everything in math is proven from those axioms and the rules of deduction (carried over from logic). A proof is a sequence of statements, where each statement is either an axiom or follows from two previous statements by a rule of deduction. Something is provable if there is a proof for it and not provable if there is not.

What Godel showed is how to investigate the notion of proof algebraically, finding common properties of all proofs and thereby showing that there are statements that are independent: that they cannot be proved, NOR can they be proved false.

Furthermore, he did this in a very general context that showed whenever the axioms are enough to talk about numbers, such independent statements can be found. Simply adding in more axioms doesn't fix this issue.

The specific questions that cannot be proved nor disproved tend to be technical. This is one reason why such statements won't impact the models used in science: the technicalities are such that experimental observations will never be able to resolve those issues.

One that is very commonly mentioned is the Continuum Hypothesis. This deals with the cardinalities of subsets of the set of real numbers (decimal numbers) and asks if there is some infinite subset that cannot be put into correspondence with either the set of natural numbers (counting numbers) nor the set of real numbers. The actual CH says that there are no such subsets.

It turns out that this cannot be proved from the standard axioms. Nor can it be shown from those axioms that there *is* a set with those properties. So CH is independent of the standard axioms. it can neither be proved nor disproved.

 

[Native]

At the time Godel worked, there was a program at the foundation on mathematics to prove it to be consistent (in other words, that it never produces a contradiction). There was another goal in this program to find a collection of axioms that was complete: it could answer any question in math (although finding the answer would still be difficult).

So, for example, it was proved that propositional logic is both consistent and complete. The same is true of predicate logic. Other extensions, like the axioms for real complete fields were shown to be complete.

This program was instituted by Hilbert, one of the best mathematicians at the time and there was a significant amount of work done towards this goal.

Godel showed that BOTH goals: completeness and a proof of consistency, are impossible.

THAT's what was shocking to those at the time. it struck at some very deep questions in what sorts of things are provable in math, which had impact on the philosophy of math.

Thanks for "cutting it out" to me And for your elaboration to @paarsurrey too.