## My take on Godel's incompleteness theorems

For the discussion of math. Duh.

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Yrreg
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### My take on Godel's incompleteness theorems

I am not a mathematician but I believe I have a good brain.

My concern is with the proof from reason that God exists, in concept as first and foremost the creator cause of everything with a beginning.

I like to learn from mathematicians here on whether I am correct with my take on Godel's incompleteness theorems.

In clear, plain, simple, concise, and precise words, I understand that Godel is telling us that any system of mathematical reasoning cannot be proven within its own system of reasoning to be true at all: so man must go outside of the mathematical reasoning in order to get to the certainty of the truth he is intent on achieving with his mathematical reasoning.

There, I will now await the mathematicians here to tell me what they see to be sense or nonsense with my take of Godel's incompleteness theorems.

gmalivuk
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### Re: My take on Godel's incompleteness theorems

A (sufficiently strong) system of logic can't prove itself consistent and in fact, if it is consistent, there must be many true statements that are not provable within the system. But another level of mathematical reasoning may be able to prove those lower level statements. You don't need to leave mathematics entirely and you definitely don't need to invoke God or anything like that.

"Yrreg cannot consistently assert this statement" is a true statement that you can't say without lying.
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
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Eebster the Great
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### Re: My take on Godel's incompleteness theorems

Also, if the system you are working with does turn out to be inconsistent, then all the statements you were trying to prove true are in fact true in that system, since every statement in an inconsistent logic is true (and false). The kind of truth you are talking about is different from mathematical truth; you want a sort of absolute truth in which some statements are simply false, and any system labelling them true is wrong in some way. That there is some perfect, correct set of axioms that is better than the others. That's not the case and certainly not what Godel proved.

Yrreg
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### Re: My take on Godel's incompleteness theorems

Thanks, dear mathematician colleagues here, for your replies.

Now, I like to invite us so far three to exchange thoughts on what is true/truth; see your posts reproduced below, with the words true/truth put with underscore by yours truly.

If I may, does what is true/truth in the minds of mathematicians not necessarily true/truth outside their minds, in the objective world of empirical reality that is independent of the human mind?

Post by gmalivuk » Sat Mar 03, 2018 10:22 pm UTC

A (sufficiently strong) system of logic can't prove itself consistent and in fact, if it is consistent, there must be many true statements that are not provable within the system. But another level of mathematical reasoning may be able to prove those lower level statements. You don't need to leave mathematics entirely and you definitely don't need to invoke God or anything like that.

"Yrreg cannot consistently assert this statement" is a true statement that you can't say without lying.
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
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Post by Eebster the Great » Sun Mar 04, 2018 1:30 am UTC

Also, if the system you are working with does turn out to be inconsistent, then all the statements you were trying to prove true are in fact true in that system, since every statement in an inconsistent logic is true (and false). The kind of truth you are talking about is different from mathematical truth; you want a sort of absolute truth in which some statements are simply false, and any system labelling them true is wrong in some way. That there is some perfect, correct set of axioms that is better than the others. That's not the case and certainly not what Godel proved.

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### Re: My take on Godel's incompleteness theorems

While true that Godel's theorems (all of them really, not even just incompleteness, his entire career!) are results about mathematical systems, this isn't exactly news to anyone.

If you'd like to talk about empirical sciences, they are up one directory and down a row. These are both pretty neat fields of inquiry.
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Yrreg
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### Re: My take on Godel's incompleteness theorems

It is with some trepidation that I come to consult mathematicians, with the assumption that they must have something useful for the advancement of my knowledge of empirical reality and truth.

That is why I am asking them, as they use the words true/truth so profusely in their writings:

Is what is true/truth with mathematicians not necessarily true/truth in the empirical world of non-mathematician mankind?

Perhaps, dear mathematician colleagues in this forum, you can do me a favor, with just referring me to a good dictionary or glossary of mathematicians' terminology.

Tub
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### Re: My take on Godel's incompleteness theorems

Yrreg wrote:Is what is true/truth with mathematicians not necessarily true/truth in the empirical world of non-mathematician mankind?

In a mathematical formal system, you start with a set of axioms, and you try to figure out everything that follows from those axioms. An axiom is just a premise, something that is assumed to be true, but cannot be proven to be true. It is perfectly valid (and often useful) to start with a set of axioms that have little to no relation to any real-world objects.

Example:

Axiom 1: 0 is a natural number
Axiom 2: Every natural number n has a unique successor (n+1) which is also a natural number

These two axioms (and a few others, but let's ignore the messy details) are enough to define the natural numbers in such a way that you can gain useful insights. You can define addition, multiplication, prime numbers and eventually cryptography based on those axioms, but you cannot prove the axioms to be true - the idea that 0 is a natural number is a convention, not a fundamental truth. 0 is not a physical entity either, so we can neither observe nor measure its existence.

Another example:

Axiom 1: Horses are blue.
Axiom 2: Everything that's blue has two legs.

The mathematician will now proclaim that horses have two legs. That's a logical conclusion from the given axioms, even if real-world applications seem limited. Nowhere did we state that horses are animals; within the formal system, "horse" is just a name.

Both sets of axioms lead to the discovery of truths in the given formal system. Both sets of axioms are consistent. But the axioms themselves are neither true nor false.

ucim
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### Re: My take on Godel's incompleteness theorems

Tub wrote:Another example:
Axiom 1: Horses are blue.
[...]

Taken as a mathematical axiom (the starting point from which we will see what interesting things arise), Axiom 1 is of course "true"... by definition. And "horse" and "blue" are either undefined terms, or words which have been defined (ultimately) in terms of undefined terms. Other examples are "point" and "line".

But, taken as a statement about the actual world around us, the statement is different. It's not an axiom, it's a premise, or a theory. Here, "horse" refers to a certain class of animals, "blue" is a certain range of colors, and the theory is that horses are blue. It would need to be clarified as to whether all or some is meant, but in either case, it can be tested by observation. Look at horses. Record their colors.

In math, there's no "observation" to make.

That's the difference.

Jose
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### Re: My take on Godel's incompleteness theorems

Maths is mostly a game. It merely says 'if X were true, what would also necessarily be true?'. Another mathematician might say 'if X were not true, what then would necessarily be true?'. Neither has any innate claim to superiority; It's all a giant game of 'what if?'

What turns it from being merely an academic exercise into something with real use is when we make observations, as ucim says. If one chain of logic says 'if A, B and C were true then X, Y and Z would follow', and another says 'if D, E and F were true then U, V and W would follow', and, Holy Cow, X, Y and Z are observed in nature and U, V and W are not, then, guess what, we might have a useful theory of nature! (aka physics)

Does that mean A, B and C are true? Maybe, maybe not. Actually, the chances are slim. The history of science is of continual refinement, with on the one hand, observations that were thought to be exactly X, Y and Z actually not being quite so on closer inspection, and, on the other, people noting 'hey, we don't actually need A, B and C to be true to get X, Y and Z; we can simplify/generalise to F and G being true instead and get the same outcome, and in addition explain other observations too.'

Newtonian physics is accurate enough to land a man on the moon; However, it's not accurate enough to predict the observed orbit of Mercury. The maths within Newtonian physics is 'correct' (self-consistent); The maths within Relativity is correct too. Which set of maths we end up using tends to depend on what we need to use it for.

Yrreg
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### Re: My take on Godel's incompleteness theorems

The way I understand, dear mathematician colleagues here, instead of the words true/truth as used in your writing on mathematical matters, If I may: the best word to use instead of true/truth, is assumption.

Outside of mathematics, the words true/truth refer ultimately to the experience of mankind in life outside and independent of his brain, or more in particular outside of game-playing inside his mind.

Thus it is true and it is a truth that woman gives birth to man, because that is the common all the time experience of mankind, in the empirical world of mankind, outside and independent of his brain, or in more particular his game-playing inside his mind.

So, I ask our mathematician colleagues here, why have you not been using the word assumption, which is the most clear, plain, simple, concise, and precise word to use for your purposes, than the words true/truth?

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### Re: My take on Godel's incompleteness theorems

A conditional truth is still a truth, even in the colloquial sense.

Mathematical truths are akin to the statement: "[Given that 'bachelor' means 'unmarried man'], if Bob is a bachelor then Bob is unmarried." The part I put in square brackets is like the axioms of a mathematical system: you're laying out what the words you're about to use mean, and you can say that they mean whatever the want to mean for the purposes of whatever you're about to say. Then, given that meanings of those words, you say something that cannot help but be true. Casually, we'd say that "all bachelors are unmarried" is a necessary truth, something that must be true of all bachelors no matter what, but really, it's only true given a certain meaning of the word "bachelor".

But it cannot help but be true that given that meaning all bachelors are unmarried, whether or not there actually are any bachelors, or any such thing as marriage, or any men, or anything at all in the universe, and even whether or not the word "bachelor" actually means that in any other context. If "bachelor" means "unmarried man", then anything that is a bachelor is unmarried, and also is a man, whatever "married" and "man" should turn out to mean, which we haven't explicitly declared here.

Mathematics is basically about very rigorous language, where language is in turn about encoding thoughts, so mathematical truths transitively turn out to be about the boundaries of what is even thinkable in principle. Everything experienceable must be imaginable in principle (what would it mean to experience something that you can't imagine? You're looking at some event and... you can't picture what that event you're looking at right now would look like?), and imagination is a kind of thought, so boundaries on thought are also boundaries on experience.

So to take our bachelor example above again: the experiential import of that is that, whatever it is you call a bachelor, if by that you mean a man (whatever you mean by that) who is unmarried (whatever you mean by that), then you will never experience a bachelor who is married. You may not experience any bachelors at all, but if you do, then they shall be unmarried. Mathematical truths 'predict' experience in the same way: given the meanings of certain words, and however those words connect to experiential phenomena, you will never experience a phenomenon that defies a mathematical theorem that derives from axioms encoding the meaning of those words.
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ucim
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### Re: My take on Godel's incompleteness theorems

Yrreg wrote:Thus it is true and it is a truth that woman gives birth to man, because that is the common all the time experience of mankind, in the empirical world of mankind, outside and independent of his brain, or in more particular his game-playing inside his mind.
No, that is not why it is true. That is why we believe it to be true. The actual truth of the matter is independent of what we believe. It is a property of the physical universe we live in, and we can discern it (at least approximately) via experiment.
Yrreg wrote:So, I ask our mathematician colleagues here, why have you not been using the word assumption [for mathematical truths]?
Because that would be the wrong word. An assumption is something we pretend is true, for the sake of argument. A (mathematical) premise is an assumption. Given the premises we have put forth, we can then draw conclusions based on syllogistic logic. Those conclusions (barring error) must absolutely follow. Those conclusions are no longer assumptions - they are truths (based on the assumptions). That is to say, they are truths within the system. All mathematical truths are only truths "within the system" for which they have been drawn; this is understood by mathematicians, and is omitted. Usually this is not a problem, but occasionally it can lead to misunderstanding, and sometimes even leads to new mathematics.

The dispute over the axiom of choice is such a case, as is non-euclidean geometry. In the first case, the axiom of choice (roughly: "out of an infinite set of sets, it is possible to select an element from each set") seems so sensible that it is accepted as a premise - an axiom. It's accepted as "true" (though this is a bit of a misnomer, as axioms are not "true"; they are "assumed", since they define the system from which we wish to draw interesting truths). However, this led to some hard-to-swallow conclusions - for example, it is possible to take a ball (the infinite set of points in space within a certain distance of another point called the center), break it into a finite number of pieces, and reassemble them into two identical balls, only by rotating them and moving them, without distorting them. So... hmmmmm.

It turns out that the axiom of choice is not actually a contradiction; a consistent set of mathematics can result from accepting this axiom. However, a different consistent set of mathematics results from rejecting this axiom. If the axiom is rejected, then it is not possible to duplicate the ball as described.

So, is the statement "A ball can be duplicated [thusly]" true? Yes, in one set of mathematics (ZFC), and no in the other (ZF). (Mathematical) truth is relative in the sense that it depends on the axioms. But given a set of axioms, it is absolute. The conclusions drawn, given a set of assumptions, are not themselves assumptions.

Jose
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Yrreg
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### Re: My take on Godel's incompleteness theorems

Pfhorrest wrote:[Bolding by Yrreg]

[ . . . . ]

Mathematical truths are akin to the statement: "[Given that 'bachelor' means 'unmarried man'], if Bob is a bachelor then Bob is unmarried." [ Etc.]

Of course, do not forget that there are bachelors in the objective world of empirical reality outside and independent of mathematicians' terminology.

Everyone conversant with English knows that bachelors are men who have not gotten married yet, in the concrete society of humans, the society that is outside and independent in the objective world of empirical reality, that has nothing to do with mathematicians' terminology.

Consider this mathematical statement: "Let x be [say] a unicorn," isn't that identical to saying, "Let us assume that x is [say] a unicorn"?

And the conclusion is "Wherefore x has one horn."

That is what I mean when I say that true/truth in mathematics or with mathematicians is an assumption, whereas bachelors are not an assumption: because there are bachelors in the objective world of empirical reality, for example, in a human society.

Mankind knows it to be true or a truth ultimately because mankind experiences that bachelors have not gotten married yet: it is not any assumption [not founded on any experience of mankind] at all that bachelors in society have not gotten married yet.

Yrreg
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### Re: My take on Godel's incompleteness theorems

ucim wrote:
Yrreg wrote:Thus it is true and it is a truth that woman gives birth to man, because that is the common all the time experience of mankind, in the empirical world of mankind, outside and independent of his brain, or in more particular his game-playing inside his mind.

[Enumeration courtesy of Yrreg]

(1)No, that is not why it is true. That is why we believe it to be true. (2)The actual truth of the matter is independent of what we believe. . . .

[Etc.]

I notice that your statement (1) and your statement (2) are not mutually coherent and not mutually consistent.

Suppose you simply just tell me whether you have ever come to witness the birth of a baby from a woman, or hear about it in your own extended family.

ucim
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### Re: My take on Godel's incompleteness theorems

Yrreg wrote:I notice that your statement (1) and your statement (2) are not mutually coherent and not mutually consistent.
Huh? I understand "coherent", but what does "mutually coherent" mean?

I baked a pie. Did I bake a pie? What do you believe? What would convince you? Before you were convinced, did I bake a pie? If you are wrong about whether or not I baked a pie, does that change the fact of the matter?

Yrreg wrote:Suppose you simply just tell me whether you have ever come to witness the birth of a baby from a woman, or hear about it in your own extended family.
Suppose you convince me of its relevance. Maybe some babies come from unicorns but we just haven't seen it happen. Maybe some babies are grown in labs. (That may actually become the case one day... probably sooner than we can deal with it.)

You may be convinced of some aspect of the real world (such as the earth being round), but your conviction does not make it true. Others are convinced it's flat. So, maybe it's half-round? We make observations and test them against the various theories, and become convinced of one or the other (or a third), but that is not why it's true. It would be true even if there were no people to argue about it.

And that's what I'm saying.
Yrreg wrote:it is true and it is a truth [...], because that is the [...] experience of mankind...
No. The "common experience" (whatever that is) is not a reason something is true. It's merely what convinces people that it's true. And in any case, this applies to the real world. It has nothing to do with mathematical truth.

Yanno, sometimes words have more than one meaning. Context clues you in as to which is meant.

Yrreg wrote:And the conclusion is "Wherefore Therefore x has one horn." (FTFY)

That is what I mean when I say that true/truth in mathematics or with mathematicians is an assumption, whereas bachelors are not an assumption:
I see what you mean, but it's wrong.
Assume {something}
Apply {reasoning}
Conclude {something else}

The conclusion is not an assumption. Using the word "assumption" to include "conclusion" makes the word "assumption" less useful in this context; as it is being used to separate the idea one begins with from the idea one ends with.

The difference I think you are elucidating is that math is abstract, and science deals with the concrete. Science is attempting to explain observations of the real world (the sun shines in the day, but hides underneath us at night). There is a sun out there, and we're trying to figure out how this all works. In the case of math, "let every integer have a successor" is an assumption, as integers are not concrete things - they are ideas. Math is a game. The "let" (or "assume") statement describes the game we're going to play. We can conclude "There are an infinite number of integers"; that is not an assumption, but it is still a statement about something abstract, rather than something in the real world.

I suspect you are using "True" and "False" to mean "corresponds with the real world" and "doesn't correspond to the real world". In that case, "true" and "false" would not be applicable to math. But mathematicians need words to mean "logically follows without contradiction" and "doesn't follow logically, and/or generates a contradiction". The words they chose are "true" and "false".

The word "savory" means "tasty". However, it also means "not sweet". Context gives you the clue.
The word "high" means "in a pleasant mentally altered state". But it also means "at a significant altitude". Context gives you the clue.

Jose
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### Re: My take on Godel's incompleteness theorems

In logic or math, roughly speaking, "true" means "follows from axioms by valid reasoning," and "false" means "has a negation that follows from axioms by valid reasoning."[1] This isn't something we started doing recently; the words (or their translations) have been used that way at least since presocratic Greek philosophy.

We say that mathematics and logic are formal sciences because they investigate only the form of statements, not their content. It doesn't matter what you are trying to enumerate; as long as their numerosity obeys the Peano axioms, we can conclude that 2 + 2 = 4 according to whatever assignment of "2" and "4" and "+" and "=" we are using that obeys said axioms. It is formally true whether or not the underlying claim corresponds to something actually real in the universe.

That is, in fact, the whole advantage of math. We can use the same theorems in many different circumstances, as long as we can write statements that have the correct form. That is also what posters mean when they say it is abstract.

[1] This is not exactly what the words mean. For instance, the Continuum Hypothesis would be neither true nor false in ZFC by the above definition, since it is independent of ZFC, but in fact, it must be either true or false by the Law of the Excluded Middle. Its truth or falsity simply cannot be determined in ZFC.

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### Re: My take on Godel's incompleteness theorems

Yrreg wrote:
Pfhorrest wrote:[Bolding by Yrreg]

[ . . . . ]

Mathematical truths are akin to the statement: "[Given that 'bachelor' means 'unmarried man'], if Bob is a bachelor then Bob is unmarried." [ Etc.]

Of course, do not forget that there are bachelors in the objective world of empirical reality outside and independent of mathematicians' terminology.

Everyone conversant with English knows that bachelors are men who have not gotten married yet, in the concrete society of humans, the society that is outside and independent in the objective world of empirical reality, that has nothing to do with mathematicians' terminology.

Consider this mathematical statement: "Let x be [say] a unicorn," isn't that identical to saying, "Let us assume that x is [say] a unicorn"?

And the conclusion is "Wherefore x has one horn."

That is what I mean when I say that true/truth in mathematics or with mathematicians is an assumption, whereas bachelors are not an assumption: because there are bachelors in the objective world of empirical reality, for example, in a human society.

Mankind knows it to be true or a truth ultimately because mankind experiences that bachelors have not gotten married yet: it is not any assumption [not founded on any experience of mankind] at all that bachelors in society have not gotten married yet.

I think it's much more useful to think about mathematics in terms of a game, as others have suggested, that happens to, sometimes, have results that are mappable to reality. This mapping, though is purely coincidental. It is perfectly possible to describe a mathematics that does not map to reality, but is still consistent and coherent within itself.

Instead, think of the game of chess. There are rules, then there are properties that follow logically from these rules, and then there is some result. If I gave you a chess board ask "Is this position checkmate?", then there is a definite true or false value that can answer that question, within the confines of the rules of chess, which I'm implicitly assuming since otherwise there is no meaningful answer to the question. Likewise, just because there is a piece in chess called a "knight" does not make chess any more real because there are actual "knights" in the world; the name is just a shorthand for a concept of a piece that moves in particular way on a chess board, and if you changed the name to an apple, bigfoot, or wogglewabble, it wouldn't change the underlying framework.

This is why when you learn formal logic, you usually don't actually deal with statements like "Let x be a unicorn"; you just deal with statements like
"Given A -> B, ~C -> ~B, Show that A -> C." This abstraction can be applied to any set of logical statements A, B, and C that have the desired properties, the details of which aren't actually relevant to the logic itself.

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### Re: My take on Godel's incompleteness theorems

To underscore LaserGuys's point, I'm going to repeat my earlier post with all relevant words replaced with placeholder letters:

Mathematical truths are akin to the statement: "[Given that 'B' means 'U and M'], if x is B then x is U." The part I put in square brackets is like the axioms of a mathematical system: you're laying out what the words you're about to use mean, and you can say that they mean whatever the want to mean for the purposes of whatever you're about to say. Then, given that meanings of those words, you say something that cannot help but be true. Casually, we'd say that "all B are U" is a necessary truth, something that must be true of all B no matter what, but really, it's only true given a certain meaning of the word "B".

But it cannot help but be true that given that meaning all B are U, whether or not there actually are any B, or any such thing as U, or any M, or anything at all in the universe, and even whether or not the word "B" actually means that in any other context. If "B" means "U and M", then anything that is B is U, and also is M, whatever "U" and "M" should turn out to mean, which we haven't explicitly declared here.

Mathematics is basically about very rigorous language, where language is in turn about encoding thoughts, so mathematical truths transitively turn out to be about the boundaries of what is even thinkable in principle. Everything experienceable must be imaginable in principle (what would it mean to experience something that you can't imagine? You're looking at some event and... you can't picture what that event you're looking at right now would look like?), and imagination is a kind of thought, so boundaries on thought are also boundaries on experience.

So to take our B example above again: the experiential import of that is that, whatever it is you call a B, if by that you mean a M (whatever you mean by that) who is U (whatever you mean by that), then you will never experience a B who is not U. You may not experience any B at all, but if you do, then they shall be U. Mathematical truths 'predict' experience in the same way: given the meanings of certain words, and however those words connect to experiential phenomena, you will never experience a phenomenon that defies a mathematical theorem that derives from axioms encoding the meaning of those words.

It just so happens that all this talk about B and U and M describes the relationship between bachelors, unmarried people, and men. But it also describes the relationship between a lot of other things: zongles are a thing I just made up, that are defined as wonky ewoks, and given that definition, it is certain that you will never see a zongle who is not wonky. You might say that that's because there's no such thing as ewoks in real life so you'll never meet a zongle at all, but even if there were, you would never meet a zongle who was not wonky, because wonkiness is part of the definition of a zongle. And even though no ewoks actually exist, and so that there are no zongles, it's still true that all zongles (all zero of them) are wonky.

In this kind of case, it's really clear to see that being B entails being U, being a zongle entails being wonky, and being a bachelor entails being unmarried. But there are a lot of complicated relationships of concepts where it's much harder to show that one sort of thing implies another sort of thing. Those long demonstrations that one thing implies another are what mathematical proofs are. And their truth is all fundamentally of the same nature as this: given certain definitions of things, there are some things about those things that cannot help but be true, no matter what does or doesn't exist.
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Yrreg
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### Re: My take on Godel's incompleteness theorems

Thanks, dear mathematician colleagues here, for your information on what mathematics is all about.

Your repetition that it is a game, like as one of you puts it, like a game of chess, that is very intriguing for us all to think about.

I can think of a game like with chess, that it is played by two human parties all inside their brain.

Is that what mathematicians want to tell mankind, that mathematics is all inside their brain, like a chess game is all inside the brain of the two human players?

Except that chess players don't use the words true/truth, but mathematicians do.

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### Re: My take on Godel's incompleteness theorems

Chess has valid moves. It has valid board positions that can arise from any given 'start', whether that is all the pieces in their initial ranks, some intermediate playable position on the way to check-/stale-mate or a contrived game position created only for the entertainment of exploring (say) three queens (and king?) vs six knights (and king?) without worrying about how the various pawn-promotions evolved into that and only how it now evolves.

And then there's further 'playful' adjustments, such as the various Fairy pieces (in conjunction with that third catagory, or to change a 'proper' game) and alternate boards (supporting >2 players, and/or >2 'dimensions', etc).

Each form of Chess is a system distinct from each other. The current 'normal' game is different from the standards used up until relatively recently. Even the 'off board' details such as the relative points assigned to pieces (guiding the logical choices of what 'ought' to be sacrificed, in case the Golden Snitch King is not actually trapped).

Chess has 'truth', in that given A, B and C, it can be shown that D is or is not True/Valid as a possible destination, under a particular (maybe standard, maybe not) game condition. (And even if Dx is more desirable than Dy. Though that's more often done by apprroximation rather than full rigor.)

Now, I've forgotten most of the Godel (also my GEB:EGB, which I started reading* two years ago, but still has the bookmark only a quarter of the way through, this third or fourth time, and dust on top of it) and mathematical logic that I learnt, so I'm not really the one to tell you how there's no "except" in the maths vs chess comparison, not the way you put it, but I think there's probably something to be learnt from similarities that you're missing.

edit: * - crucial mistype! That's rereading. As in I have read it cover-to-cover several times in the last tumpty-tump years, but the last time I started it I have not yet finished it, having too many other books on the go and other distractions.
Last edited by Soupspoon on Fri Mar 16, 2018 2:49 pm UTC, edited 1 time in total.

Eebster the Great
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### Re: My take on Godel's incompleteness theorems

Analogies aside, much of mathematics is literally recreational and often involves looking at actual games. It's still perfectly valid mathematics, but it is difficult to apply in practice. For instance, solving the game of checkers is interesting, but it's not really useful.

You are specifically focusing on the parts of math that do have practical application and demanding to know why. That's a deep question (maybe), but it's a philosophical one, not a mathematical one. The mathematical truth of a theorem is not contingent on its relevance to anything.

ucim
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### Re: My take on Godel's incompleteness theorems

Math is a game. The object of the game is to figure out the rules.

That is... to figure out which sets of rules are interesting (lead to interesting results).

On the other hand, the universe is a thing. We presume that it operates based on a set of rules, and seek to figure which rules are in the set. However, this might be an incorrect presumption. It's an act of faith - faith that is strengthened every time we figure out another one of the rules it seems to work by. If it works by rules, then math can be used to understand the universe, because math uses rules too.

That's the difference.

Jose
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### Re: My take on Godel's incompleteness theorems

Here's a stab at answering that philosophical question:

Mathematics as a field is essentially exploring the abstract space of possibilities. Whatever happens to actually exist will, of course, be possible. It should thus be entirely unsurprising that some mathematical structures describe the things that actually exist. I actually find the "unreasonable effectiveness of mathematics in the natural sciences" entirely trivial and an odd thing to find surprising. What else would you possible expect? That the things described by natural sciences would be... undescribable? Completely incoherent? Absolutely anything possible can be described by some mathematical structure or another, whether or not we've found such a structure or explored its implications yet. And given that we live in the actual world, it's entirely unsurprising that the kind of structures we've explored rigorously are the kinds exhibited in that world.
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Eebster the Great
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### Re: My take on Godel's incompleteness theorems

I tend to agree. Moreover, some basic concepts, like the natural numbers, are so broadly applicable anyway, that it's hard to imagine how they could fall to be useful.

gmalivuk
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### Re: My take on Godel's incompleteness theorems

Yeah the alternative to a universe we can describe with math is a universe that is internally inconsistent and inherently unpredictable.

We assume it's not like that for the sake of being able to do anything at all.

If it's predictable, then we can use math to figure out how to make predictions about it. If it's not, then nothing we do has any consistent effect and therefore doesn't matter in any possible sense of the word.

And if the universe is controlled by a god, the same point applies: either that god is analyzable with mathematics, or nothing matters or makes sense.
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Yrreg
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### Re: My take on Godel's incompleteness theorems

gmalivuk wrote:Yeah the alternative to a universe we can describe with math is a universe that is internally inconsistent and inherently unpredictable.

We assume it's not like that for the sake of being able to do anything at all.

If it's predictable, then we can use math to figure out how to make predictions about it. If it's not, then nothing we do has any consistent effect and therefore doesn't matter in any possible sense of the word.

And if the universe is controlled by a god, the same point applies: either that god is analyzable with mathematics, or nothing matters or makes sense.

The way I see man's exertion to know the existence of things in the totality of reality at all, it is that we must keep in mind that there is a distinction between the world we have in our mind and the objective world of empirical reality outside and independent of our mind.

Now, man notices that the world he makes of existence inside his mind is not all the time corresponding to the objective existence i.e. empirical reality outside and independent of his mind.

Mathematics is one kind of a mind world; and the way I see it, mathematicians claim to know this world and claim that it is superior to the world of empirical reality outside and independent of his mathematical world inside his mind.

With Godel's incompleteness theorems, he sees himself to have come to the certainty of the mathematical world inside mathematicians' mind to be founded on what I call with the term i.e. English noun word in plural number, arbitrarieties: and here is the in a way should be distressing to mathematicians, the world of mathematics in the mind of mathematicians is inherently shaky, opposite to solid and firm.

And no matter that mathematicians want to use words from the empirical world outside and independent of their mind, like the words true/truth, in order to continue to dwell inside their mind with intellectual security, with the use of the words true/truth to describe their mathematical things and ideas, that mathematical world is still just the same inherently shaky, i.e. not as true and grounded on truths as the objective world of empirical reality is.

Proof of that is the fact that mathematicians can live inside their mind to occupy themselves with mathematical things and ideas, nonetheless they are subject to the objective world of empirical reality outside and independent of their mind, at the risk of compromising their empirical health and even incurring termination of their biological existence, and thereby the end of their mathematical world with the extinction of their brain.

For example, mathematicians have got to breathe air to stay alive, to ingest food, to defecate, to urinate, to avoid falling off a cliff, no matter that they can prove to themselves that they are on the basis of their mathematical world inside their mind, that they are immune to all hazards of death of the physical empirical world of biological existence: they will extinguish themselves and thereby also their mathematical world.

So?

So mathematicians must always be cautious that they don't get themselves hit by a speeding car as they dwell inside their mathematal world of a mathematicized mind.

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### Re: My take on Godel's incompleteness theorems

Yrreg wrote:The way I see man's exertion to know the existence of things in the totality of reality at all, it is that we must keep in mind that there is a distinction between the world we have in our mind and the objective world of empirical reality outside and independent of our mind.

But the way you see it makes no sense and does not resemble what anyone in this thread has been saying. Math is not a "mind world," just the study of formal statements. It is not a "world" or a "reality," just a method. And nobody thinks math is "superior" to reality, whatever that might mean.

ucim
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### Re: My take on Godel's incompleteness theorems

Yrreg wrote:mathematicians claim [...] that it is superior to the world of empirical reality

No. Just....no.

Yrreg wrote:And no matter that mathematicians want to use words from the empirical world outside and independent of their mind, like the words true/truth, in order to continue to dwell
These are not "holy words". There's nothing magical or mystical or powerful about the words "true" and "false" (or their derivatives). They are just words. Ordinary words. Mathematicians needed an utterance to describe the concept of "following from the axioms in question", and they used words which already carried that shade of meaning. In adapting them to mathematics, they were given a specific meaning within mathematics, just like any other word that has a specific meaning in any other context.
Yrreg wrote:...the world of mathematics in the mind of mathematicians is inherently shaky...
No, it's not "shaky", it's "limited". GIT says (roughly) that there are true statements (about a domain) that cannot be proven (from within that domain). It was previously thought that all true statements could be proven within their domain.

This does not invalidate any proofs that already exist, what it does do is show that some hoped-for proofs cannot exist, even if the thing they would prove is actually true. There are true statements that have no proof; we just don't (and can't!) know which ones they are.

Yrreg wrote:So mathematicians must always be cautious that they don't get themselves hit by a speeding car as they dwell inside their mathematal world of a mathematicized mind.
The same is true of politicians, cartoonists, and beach bunnies.

In any case, so what? I don't see what you're getting at any more.

Jose
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### Re: My take on Godel's incompleteness theorems

Yrreg wrote:[...] and here is the in a way should be distressing to mathematicians, the world of mathematics in the mind of mathematicians is inherently shaky, opposite to solid and firm.

And no matter that mathematicians want to use words from the empirical world outside and independent of their mind, like the words true/truth, in order to continue to dwell inside their mind with intellectual security, with the use of the words true/truth to describe their mathematical things and ideas, that mathematical world is still just the same inherently shaky, i.e. not as true and grounded on truths as the objective world of empirical reality is.

Treatid? Is that you?

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### Re: My take on Godel's incompleteness theorems

I definitely checked when I saw where this thread was going. (It doesn't seem to be.)
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Yrreg
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### Re: My take on Godel's incompleteness theorems

Dear mathematician colleagues here, thanks a lot for your replies to my posts, I have come to insights on the nature of mathematics.

"[There is] The Unreasonable Effectiveness of Mathematics in the Natural Sciences."

It is the title of an article published in 1960 by the mathematician-physicist Eugene Wigner.

My take on the unreasonable effectiveness is because it will occur that by trial and error which is endemic in mathematics, sooner or later a connection comes forth from mathematicians, which gives an explanation in the mind of man for a physical phenomenon in the empirical world of nature outside and independent of the mind of man, including the mind of mathematicians.

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### Re: My take on Godel's incompleteness theorems

For my part, having not looked deeply into the work, I see it as an analogue of one form or other of the Anthropic Principle.

If (to use an example) the calculations derived from bodies moving ballistically on Earth did not then go on to explain the previously-observed-but-not-understood orbital mechanics, we would not now be using the same equations.

Indeed, with Mercury's deviation due to borderline relativistic effects (described well, it seems) and the problem of galactic rotations (still at the stage of having multiple explanations jockeying for position) obviously show that there is less "unreasonable effectiveness" of the basic theory of gravitation than one might perhaps casually assume in a cart-before-the-horse assertion.

How one proceeds from there largely depends upon which type of Anthropic argument one subscribes to, or which way one normally slices with Occam's razor in various arguments that I see not too far from the current line of questioning. At face value, though, I say that my hasty reading of the "unreasonableness" thing, which may mean I missed the real message, is that it is made with the same sort of hubris as the pre-Einsteinian quote that basically "all science is now known (barring the details)" when we should know that it is an ongoing process of refinement of understanding (Sagan: "We are a way for the Cosmos to know itself"…?) and "trial and error" does also make it sound more like a Stock Tip Fraud method rather than a more honest (yet still imperfect) honest assessment of future market conditions.

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### Re: My take on Godel's incompleteness theorems

Maths is "unreasonably effective" in the sense that there's no reason the universe couldn't be unpredictable on a macro level. However, as Soupspoon and others have said, in such a universe intelligent life (and perhaps all life) would not have evolved.

For example, I read that a universe with 2+ time dimensions would not have effect following cause in any meaningful way and so an intelligence capable of developing maths would not arise.

Because we are here, the universe is therefore rational to a very great degree, and, since maths is general enough to model all possible rational systems, it can model ours.

That doesn't mean the rug couldn't get swept out from under our feet: Maybe we're inside a simulation and someone's just about to change a load of parameters.

Physics is empirical not objective: It can predict the future but only by assuming it will behave like the past. You seem to think that mathematicians have some kind of hubris here but they really don't; They just assume the future will behave like the past because if that's not true then nothing and noone can meaningfully predict the future.

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### Re: My take on Godel's incompleteness theorems

From Elasto
"Physics is empirical not objective: It can predict the future but only by assuming it will behave like the past. You seem to think that mathematicians have some kind of hubris here but they really don't; They just assume the future will behave like the past because if that's not true then nothing and noone can meaningfully predict the future."

Is there a way at all for man to be objective but away from empirical experience altogether?

I seem to have in my stock knowledge that Plato is the patron saint of mathematicians, in that he holds there is a world that is not founded ultimately on man's experience of the world, and it is accessed with man's mind - the world of mathematics.

It seems he never takes notice of the fact that as he breathes and is experiencing his breathing, then he is alive and has access to the mathematical world all in his mind,

When Plato starts experiencing the further and further loss of his breath, he realizes that he is going out of life, and wherefore with his demise, so also his whole mathematics world in his mind.

And mathematicians today do a lot of fiction with mathematics, then they wait for some event to turn up, and/or for man to come to the discovery of some event in the world, by which event he has found the proof for the fiction he builds with mathematics in his mind.

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### Re: My take on Godel's incompleteness theorems

Yrreg wrote:Is there a way at all for man to be objective but away from empirical experience altogether?

I seem to have in my stock knowledge that Plato is the patron saint of mathematicians in that he holds there is a world that is not founded ultimately on man's experience of the world, and it is accessed with man's mind - the world of mathematics.

Well, mathematicians don't have saints. I guess you're trying to paint mathematics as a religion but it really couldn't be more polar opposite.

While you could describe mathematics as a world, it's obviously of a fundamentally different kind to the real world. So, yes, it's possible to be objective about something you control in its totality and is eternal and unchanging, like an abstract mathematical theorem, but by definition an individual can ultimately only ever have subjective knowledge of the real world through qualia.

And mathematicians today do a lot of fiction with mathematics, then they wait for some event to turn up, and/or for man to come to the discovery of some event in the world, by which event he has found the proof for the fiction he builds with mathematics in his mind.

Once again you seem to be conflating mathematicians with scientists.

That nitpick aside, you seem to fundamentally misunderstand the scientific method: Scientific theorems can never be proved, they can only be disproved. Sure, you can amass a billion data points that agree with your model, but the billion-and-oneth might turn out to disprove it. All good scientists know this. Moreover, there are an infinite number of distinct mathematical models that match the finite number of observations humanity might make.

And yet, despite all this, science has created the miraculous devices that permit you to transfer thoughts from your brain to the rest of the world virtually instantaneously, so it must be doing something right, right...?

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### Re: My take on Godel's incompleteness theorems

Mathematical Platonists are definitely a minority, but yes, they do claim mathematical objects are objectively real things in some sense. I don't really get it, personally.

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### Re: My take on Godel's incompleteness theorems

Eebster the Great wrote:Mathematical Platonists are definitely a minority, but yes, they do claim mathematical objects are objectively real things in some sense. I don't really get it, personally.
Don't be so square, man. You're practically a regular hexahedron!

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### Re: My take on Godel's incompleteness theorems

Eebster the Great wrote:Mathematical Platonists are definitely a minority, but yes, they do claim mathematical objects are objectively real things in some sense. I don't really get it, personally.

As I say, I can see how a mathematical theorem can be objectively true in the sense that any rational being anywhere would come to the same conclusions given the same assumptions, so the definition of a circle/sphere/etc. is in some sense an objective truth, but I don't think that means they think there is a literal place where true circles/spheres/etc. exist.

(If they do, they have indeed strayed from maths into religion, but mathematicians are just people after all; Noone is perfectly rational, and it'd probably be a rather cold, unfeeling world if everyone were!)

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### Re: My take on Godel's incompleteness theorems

Eebster didn't say objectively true, but objectively real.

And Platonists haven't left math for religion, so much as for philosophy, which is exactly where you should be if you're discussing the ontological status of mathematical objects.
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elasto
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### Re: My take on Godel's incompleteness theorems

gmalivuk wrote:Eebster didn't say objectively true, but objectively real.

And, as I say, with caveats I can agree with 'objectively true' but not 'objectively real'.

And Platonists haven't left math for religion, so much as for philosophy

If Platonists are arguing over objective truth, then, yes, that's philosophy. But if they are claiming that a perfect sphere is objectively real somewhere, I'd argue that's more of a religious position. That's seriously splitting hairs though as the line between philosophy and religion is pretty blurry.

(It's especially splitting hairs in the sense that, in a way, religion is a subset of philosophy, as is everything else, especially maths.)