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merjet

Math and reality

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I made the following comment on the Charles Tew thread:

In another video he talks about mathematics. I don't remember which one. He prefaced his remarks with his not being a mathematician, but what he said about math did not sit well with me. As I recall, he said mathematics is about reality. Yes and no. It is also about our concepts. Show me a matrix, differential equation, integral or complex number in reality that wasn't written by some human being, then I will reconsider.” Link.

MisterSwig suggested privately that I say more about this in a separate thread. Before I do, what was it that Charles Tew said that did not sit well with me? The video is ‘Sam Harris Doesn't Understand Math‘ (link).  Tew launched a tirade on Sam Harris' saying that the Probability{Jesus will come back in Jackson County, Mo.} < Probability{Jesus will come back somewhere}. Defending that claim when the interviewer challenged him, Harris said it is a mathematically precise statement.

In addition to the title he gave the video, Tew made several other comments, including the following:

- Harris is disastrously wrong about his Jesus claim.

- He said that mathematics is about the world. It applies to reality.

- He claimed Harris had a Platonic understanding of math.

- Mathematicians don’t know what they are talking about, because they aren’t philosophers. Ditto for physicists.

Sam Harris was a little imprecise. He should have said, simplifying, Pr{Jesus to Jackson County} <= Pr{Jesus to somewhere}. Not “less than”, but “less than or equal to.” Even holding that both probabilities are zero like Tew said, the “less than or equal to” formulation is true. Also, more generally, Pr{A} <= Pr{B} if A is a proper subset of B. That is the mathematical principle Harris appealed to, even if he didn’t say it wholly correct. In my opinion, Pew’s asserting that Harris doesn’t understand math is a gross exaggeration. Not even arithmetic and some mathematical probability?

That’s all I will say about Harris. I move on to some of Tew’s claims. First, his assertion about mathematicians and physicists is pompous and insulting to many people, quite a few I know personally. Next, is mathematics about the world? Maybe Tew took that claim from the title of the book by Robert E. Knapp. Mathematics is About the World. Nevertheless, Tew's understanding of math seems to me pretty shallow.

I have been aware of Knapp's book for a while but haven’t read it. Anyway, in my view mathematics is also about the ideas we use to describe the world quantitatively.

Let’s start with arithmetic. Is arithmetic about the world? Mostly yes, but not entirely. Consider 5 – 2 = 3. That’s true for all things countable. But what about 3 – 5 = -2? If I begin with 5 dimes in my hand and remove 2 of them, 3 dimes remain in my hand. However, beginning with 3 dimes in my hand and removing 5 of them is impossible. On the other hand, if the temperature is 3 degrees Fahrenheit or Celsius and then falls 5 degrees, saying it’s then - 2 degrees is valid and about the world. Pure math is an abstract discipline, so mathematicians usually ignore exceptions like not being able to remove 5 dimes from my hand when there are only 3 dimes there. By the way, when negative numbers were first considered, they were regarded as fictitious or false. The algebraic equation x^2 + x – 6 = 0 has two solutions (roots), x = +2 and x = -3.

Moving on, a parabola (or circle, or ellipse, or hyperbola) is based on a conic section. The algebraic equation for one can be expressed in Cartesian coordinates – invented by Descartes -- or polar coordinates. Similar for volumes such as that of a cone, cylinder, or sphere. Is such mathematics about the world? It surely is. Cartesian and polar coordinates have a direct correspondence to the real world. The axes in both can correspond to distances in the real, external world.

On the other hand, there is another coordinate system which has no such direct correspondence to the real, external world. It is often called the complex plane and is pictured here. The horizontal axis is for real numbers, but the vertical axis is for imaginary (or complex) numbers. We can’t use imaginary numbers to express distances in the real world. On the other hand, imaginary numbers are used to describe the real world in physics, more specifically quantum mechanics (link). The same complex plane coordinate system is shown there again.

When imaginary or complex numbers were first systematically explored by Euler, Gauss, and Hamilton more than 150 years ago, their practical use was unknown. So one could say that imaginary or complex numbers were not about the world then. Times have changed. A practical use of them was found many years later in quantum mechanics. So one could say that imaginary or complex numbers are about the world now.

By the way, I earlier gave an algebraic equation that had two real solutions (roots). Here is an algebraic equation that has no real number solutions (roots): x^2 + 4x + 5 = 0. The two solutions (roots) are -2 + i and -2 – i, where i is the imaginary (complex) number equal to the square root of -1.

To be continued in another post(s). I will say something about matrices, calculus, differential equations, maybe more.

Edited by merjet

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By switching 'possibility' and 'probability' over to the science of measurement, it provides a numeric value without having to address the epistemic issue of where it lies on the continuum of certainty.

The status of possibility is a recognition that some, but not much, evidence supports a hypothesis, and nothing known contradicts it.

The status of probability is a recognition that a substantial amount of evidence supports a hypothesis, and while nothing known contradicts it, certainty is not yet conclusive. 

In probability and statistics, to measure something presumes that there is something to be measured.

In essence, some areas of math use the prestige the certainty of math purports, implicitly serving as a form of camouflage obscuring a lack of a philosophic foundation. 

Pat Corvini used words to the effect that in math, if it is wrong, it is influentially wrong. She was speaking about set theory in particular. The principle is sound more broadly.

 

Edited by dream_weaver

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2 hours ago, dream_weaver said:

Pat Corvini used words to the effect that in math, if it is wrong, it is influentially wrong. She was speaking about set theory in particular. The principle is sound more broadly.

Can you elaborate? Is there an online source for reading more of what she said?

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A few years ago I purchased and listened to Pat Corvini's two sets of lectures on number:
1. Two, Three, Four and All That; and 
2. Two, Three, Four and All That: The Sequel

The main topic is her view of numbers. A lesser topic is criticism of Cantor's claims about infinite sets, and his method, with she calls postulational and contructive. Corvini does not say so, but the postulational/contructive philosophical view is epitomized by the famous mathematician David Hilbert's opinion that the most reliable way to treat mathematics is to regard it not as factual knowledge, but as a purely formal discipline that is abstract, symbolic, and without reference to meaning.

Her method focuses on "the what" of numbers, whereas Cantor's methods focuses on "the how" of numbers. She sharply distinguishes between counting -- which use only the positive integers -- and measuring -- whose domain is the real numbers (integers + rationals + irrationals). Cantor's method of one-to-one correspondence blurs the distinction.

She talks about Cantor in the 1st and 3rd lectures of The Sequel. The last 1/3rd or so of Sequel #2 and the first half or so of Sequel #3 elaborate her view of measurement. Then she returns to Cantor and the postulational/constructive view of the rational and irrrational numbers. In her view there are two sorts of infinities -- counting (conceptualized by the positive integers) and measuring (conceptualized by real numbers and attained by subdividing). The postulational/constructive method blurs the distinction and treats open-ended construction like a concrete.

I much agree with what she says, but believe there are even stronger criticisms of Cantor's nonsense. At one point in Sequel #1, Corvini talks in terms of 2-to-1 correspondence, but not any wider range of multiple-to-1 correspondences. Nor does she utilize part-whole logic to criticize Cantor's nonsense.

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Well, I just toasted my brain writing all morning, so I'll have to consider the advanced math stuff later. Thanks for doing that.

Regarding the Jesus argument, would it matter which assumptions you make about Jesus? For example, if we assume that he existed in some form, but not that he was miraculous, then it's nearly certain that he died and deteriorated, never to return anywhere in any form. But, since there is no current evidence of his death, we might accept the tiny chance that something weird happened within the realm of natural laws. For example, advanced aliens visited Earth, snatched up Jesus, stuck him in cryofreeze, and now plan to drop him in Jackson County next winter. The level of probability seems to depend on your initial assumptions.

There also seems to be a contextual issue here. Okay, I'm out. Can't think anymore. 

Edited by MisterSwig

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2 hours ago, dream_weaver said:

Your recollection is better than mine on the details.

😊 I must confess that my recollection is not as good as it might seem. My last post was copied from something I wrote a few years ago and then edited a little.

Edited by merjet

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The system of equations:

2x + 3y = 16

x + 2y = 10

can be placed in matrix form and be pictured with 2-dimensional Cartesian coordinates. I wish I could show the matrix form, but I don't know how to do so here. I omit the picture (graph), too.

Similarly, a system of 3 equations and 3 unknowns can be placed in matrix form and be pictured with 3-dimensional Cartesian coordinates.

On the other hand, a system of higher order, 4 or more, cannot be pictured with spatial coordinates of any kind. Hence, I for one would not describe such a system as "about the world", but rather "about how we can think about the world." Surely, when we start talking about multiplying matrices, we are not talking "about the world", but rather "about how we can think about the world."

Calculus, with its concepts of limits, infinite series, infinitely large and infinitely small, we are not talking "about the world", at least the external world, but rather "about how we can think about the world" and/or methodical thought that takes place in our internal, mental world.

Edited by merjet

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On 7/14/2019 at 7:09 AM, merjet said:

Let’s start with arithmetic. Is arithmetic about the world? Mostly yes, but not entirely. Consider 5 – 2 = 3. That’s true for all things countable. But what about 3 – 5 = -2? If I begin with 5 dimes in my hand and remove 2 of them, 3 dimes remain in my hand. However, beginning with 3 dimes in my hand and removing 5 of them is impossible. On the other hand, if the temperature is 3 degrees Fahrenheit or Celsius and then falls 5 degrees, saying it’s then - 2 degrees is valid and about the world.

Are we dealing with a false dichotomy here? It's not that math is about either the world or our abstractions of the world. Math is about discovering ways to measure things. And since we have imaginary things in our heads (two absent dimes, for example), we need a method of measuring them.

Edited by MisterSwig

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3 hours ago, MisterSwig said:

Are we dealing with a false dichotomy here? It's not that math is about either the world or our abstractions of the world. Math is about discovering ways to measure things. And since we have imaginary things in our heads (two absent dimes, for example), we need a method of measuring them.

This I think is a species of a more general principle.

ALL mental content is ultimately “about” living in the real world.  Of course we can think about thoughts and imagine and abstract from abstractions and go play with mental contents off in the far lands of   the mentality, but we do so with reference to its ultimate connection with reality... finally in our ability to use our mental capacity to gain knowledge of and act in the world.   Some small bit of mental content might not have a direct and concrete referent in reality, but every sane concept has to be at least indirectly connected to reality.  Even imagination is often connected to, taken from, or relatable to parts of reality.  

A concept whose referent is wholly disconnected from all of reality would be the product of insanity.

Math although highly abstract at times is ultimately about quantities and measurement of aspects of reality not an untethered game tossed upon a sea of floating abstractions.  This might remind one of Rand’s reframing and reinforcing the term morality for man living life in reality rather than a nonexistent soul’s duty to a nonexistent God.  In the same way and for the same reason Math as a value for man is about the world.

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On ‎7‎/‎14‎/‎2019 at 10:09 AM, merjet said:

First, his assertion about mathematicians and physicists is pompous and insulting to many people, quite a few I know personally

I have an MSc. in physics and not quite a minor (with my BSc. w hon in physics) in Math...  I have to say that the VAST majority of professors and students in Math and Physics have a severely Rationalistic, Platonic, and Idealistic (as in Idealism) slant.  Very few have a good grounding in valid philosophy, and are quite happy to embrace logical fallacies, engage in compartmentalization, and come with a desire to find incomprehensibility, like a strange desire not to understand, but to embrace the impossible. 

I would put the ratio at 80%-20%... and I would say only with my exposure to Objectivism have I been able to put myself in the more objective out of that 20%. 

It is unfortunate but it is not surprising.  Like people who go into philosophy with the best intentions, certain mentalities are drawn into "high math" and "high physics"... the most common baggage is a rejection of "low reality", the "concrete", the "practical", money and business (life?) and an embrace of the "ideal", the abstract, the Academy... it is hard for me to convey ... one has to live it to really understand the culture, the philosophical undercurrent.

 

 

Roger Penrose's "Three world" theory is something most Physicists and Mathematicians would not bat an eye at.  Rather than mental contents "in here" versus the world "out there", Penrose imagines a relationship between three "worlds": the "Platonic Mathematical World", the "mental world", and the "physical world".

It might be understandable that the false dichotomy between consciousness and existence is exaggerated  (minds are not supernatural but consciousness is a unique functioning/attribute/property of reality) but a separate world of Ideas for math... only comes from studying numbers to a point that they take on an existence all to themselves... reification by sheer repetition.

three-worlds-roger-penrose.jpg

Here is an article which talks about this:

https://astudentforever.wordpress.com/2015/03/13/roger-penroses-three-worlds-and-three-deep-mysteries-theory/

and another one

https://scientificgems.wordpress.com/2013/05/18/three-worlds/

 

 

Edited by StrictlyLogical

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Great mathematicians are capable of bad philosophy. Two examples are Descartes and David Hilbert.

"Mathematics is a game played according to certain simple rules with meaningless marks on paper." - David Hilbert  Link

Maybe Hilbert was trying to be funny, but there is plenty of room for doubt. Morris Kline in Mathematics: The Loss of Certainty said about Hilbert's view: "The most reliable way to treat mathematics is to regard it not as factual knowledge, but a purely formal discipline that is abstract, symbolic and without reference to meaning" (page 247). Also, note the subtitle of Knapp's book.

 

Edited by merjet

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19 hours ago, StrictlyLogical said:

Some small bit of mental content might not have a direct and concrete referent in reality, but every sane concept has to be at least indirectly connected to reality.

Okay, so is there a type of math system that's wholly insane, i.e., not even indirectly connected to reality? Algebra, for the most part, still deals with counting things. Geometry and trigonometry deal with measuring shapes. Calculus measures changes. Probability measures potentials. I understand how concepts like "negative numbers" and "imaginary planes" have no direct relation to physical reality, but even those ideas are indirectly related. Is there, however, a branch of math that has absolutely nothing to do with the outside world, and is entirely in the mind?

 

Edited by MisterSwig

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1 hour ago, MisterSwig said:

Okay, so is there a type of math system that's wholly insane, i.e., not even indirectly connected to reality? Algebra, for the most part, still deals with counting things. Geometry and trigonometry deal with measuring shapes. Calculus measures changes. Probability measures potentials. I understand how concepts like "negative numbers" and "imaginary planes" have no direct relation to physical reality, but even those ideas are indirectly related. Is there, however, a branch of math that has absolutely nothing to do with the outside world, and is entirely in the mind?

 

Here is the interesting part.

Since the fundamentals of math grows out of concepts which ultimately come from percepts, absolute disconnection is difficult to achieve.  As for entire branches of math... I do not know.

The problem is not that math can get VERY abstract, VERY far from the concretes of reality which are connected to them, but that those who are DOING the math dispense with that connection entirely.  Embracing either the idea that it is a game of the mind disconnected from reality or a revelation of a Platonic Ideal Reality risks the creation of mathematical concepts which do become disconnected in the same way floating abstractions can become disconnected.

Imagine, like some freshman philosophy students, you embrace the idea that what you do need not conform to anything, nothing in reality, none of the axioms, and you can simply create systems out of nothing and with no rules except what you give it.  WE know such creations do not fall within what is VALIDLY to respectively be called knowledge or the love and study of it (philosophy) nor mathematics.  Attempting to "Invent" a "number system" (system of symbols) which represent contradiction as part of a formal system would constitute such an insanity... I'm not sure it was ever done... but a mentality that thinks of it as "fair game" is NOT hinged to reality.

I believe there is a strong case to be made philosophically against various mathematical constructs and even well accepted "profound" conclusions.  Symbolic logic and nonsensical implication (truth tables) and Gödel's theorem are the types of things that require better Objective explication and understanding.

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1 hour ago, MisterSwig said:

Okay, so is there a type of math system that's wholly insane, i.e., not even indirectly connected to reality?

Maybe some of the non-Aristotelian "logics" that have been developed. Some of them allow true contradictions and three or four truth values.

Or you could look at Euclidean geometry vs. infinitely many non-Euclidean alternatives. It's hard to see how they could all have a connection to reality.

This isn't something I've thought a lot about, I'm just tossing out some possible examples.

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On 7/14/2019 at 2:49 PM, merjet said:

David Hilbert's opinion that the most reliable way to treat mathematics is to regard it not as factual knowledge, but as a purely formal discipline that is abstract, symbolic, and without reference to meaning.

I would like to see a direct quote of Hilbert on that. 

Hilbert did discuss that, in one way, formal systems can be viewed separately from content or meaning. But that does not imply that in another way they cannot be viewed with regard to content or meaning. Indeed, Hilbert was very much concerned with the "contentual" aspect of mathematics. 

Granted, descriptions of Hilbert as viewing mathematics as merely "a pure game of symbols", "without meaning", et. al do occur in literature that simplifies discussion of Hilbert. But for years I have asked people making the claim (here moderated to "reliability") to provide a direct quote from Hilbert. And just looking at Hilbert briefly is enough to see that he was very much concerned with the contentual in mathematics.

I'm simplifying somewhat, but Hilbert distinguished between (1) statements that can be checked by finitistic means and (2) statements that cannot be checked by finitistic means.

Finitistic means are those that can be reduced to finite counting and combination operations - even reducing to finite manipulations of "tokens" (such as stroke marks on paper if we need to concretize). This is unassailable mathematics, even for finitists and constructivists. If one denies finitistic mathematics, then what other mathematics could one possible accept?

On the other hand, mathematics also involves discussion of things such as infinite sequences (try to do even first year calculus without the notion of an infinite sequence). So Hilbert wanted to find a finitistic proof that our axiomatizations of non-finitistic mathematics are consistent. So, there would be unassailable finitistic mathematics (which has clear meaning - that of counting and finite combinatorics) and there would be axiomatized non-finitistic mathematics (of which people may disagree as to whether it has meaning and, if it does have meaning, what that meaning is) that would at least have a finitistic proof of its consistency.

So, of course Hilbert regarded finitistic mathematics as having meaning and being completely reliable. And, I'm pretty sure you will find that Hilbert also understood the scientific application of non-finitistic mathematics (such as calcululs). But he understood that it cannot be checked like finitistic mathematics; so what he wanted was a finititistic (thus utterly reliable) proof that non-finitistic mathematics is at least consistent.

However, Godel (finitistically) proved that Hilbert's hope for a finitistic consistency proof cannot be realized. 

Regarding looking at formal systems separately from content: Imagine you have a formal system such as a computer programming language. We usually regard it to have meaning, such as the actual commands it executes on physical computers or whatever. But also, we can view the mere syntax of it separately, without regard to meaning. One could ask, "Is this page of code in proper syntax? I don't need to know at this moment whether it works to do what I want it to do; I just need to know, for this moment, whether it passes the check for syntax." 

So formal symbol rules can viewed in separation from content, or they can also be viewed with regard to content. Hilbert emphasized, in certain context the separation from content, but in so doing, he did not claim that there is not also a relationship with content.

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On 7/14/2019 at 2:49 PM, merjet said:

Cantor's nonsense.

Cantor himself did not formalize his mathematics, and at certain points he had to explain paradoxes merely by invoking the notion of  "inconsistent totalities" (or whatever his exact phrase). However, there is a view that ZFC is a formalization that is true to the heart of Cantor's mathematics. In that way, for mathematics itself, it is better to address ZFC, the more modern formulation of set theory than to address Cantor. And for questions about the philosophy of mathematics, Cantor had his own views, but they do not control the actual mathematics of ZFC and there are philosophical views supporting ZFC that are not necessarily beholden to Cantor. 

ZFC is used to provide a formal axiomatization of the mathematics for the sciences. By 'formal' we mean that that there is an algorithm to verify of any purported proof whether or not it is needed a correct formal proof in the system. Now, if one wishes to reject ZFC, then there are at least these two choices.

(1) Reject the notion of providing a formal axiomatization of the mathematics for the sciences. 

(2) Provide an alternative formal axiomatization of the mathematics for the sciences. There are alternatives based in other approaches (including perhaps even ultra-finitistic), but then we would compare these with ZFC for such things as how complicated they are to understand, how closely they fit with the kinds of reasoning mathematicians ordinarily use, etc. 

 

 

Edited by GrandMinnow

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37 minutes ago, GrandMinnow said:

I would like to see a direct quote of Hilbert on that. 

 

25 minutes ago, GrandMinnow said:

Where is that found in the actual writings of Hilbert?

Maybe he didn't write it but spoke it. In any case, Kline referring to Hilbert in Mathematics: The Loss of Certainty wrote something quite similar. Maybe Hilbert did not intend it apply to all mathematics, but only part of it. As an aside, there is this.

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If he spoke it, we do not here have a specific instance.

Quite some time ago I read that SEP article you linked to; I don't wish to reread it again right now. If you like, you could mention which passage you have in mind. Indeed, as I skimmed the article, I saw that it makes mention along the lines of what I just posted. For example:

"That a set of sentences is consistent in Hilbert's sense is a matter that's entirely independent of what its geometric terms mean [...]"

Indeed, consistency (formal consistency in the sense whether a formal mathematical system does not prove both a formal sentence and its negation) is not a question of content or meaning. But that does not entail that mathematics is just a game of symbols or that mathematics is devoid of content or meaning. 

As to Kline, the same question: What is his source?

At this point, unless there is some attribution to Hilbert, it cannot be taken that he ever wrote or said that mathematics is just a symbol game with no meaning.

Edited by GrandMinnow

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5 hours ago, StrictlyLogical said:

Symbolic logic and nonsensical implication (truth tables)

[...]

and Gödel's theorem are the types of things that require better Objective explication and understanding.

Symbolic logic and truth tables are absolutely basic tools in the research and design of the computer you're using right now.

Godel's incompleteness theorem can be performed in perfectly finitistic mathematics (reducible to the manipulation of concrete tokens, such as we perform in plain arithmetic and combinatorics of natural numbers). I don't know what specifically you object to in Godel's work. 

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37 minutes ago, GrandMinnow said:

Symbolic logic and truth tables are absolutely basic tools in the research and design of the computer you're using right now.

Godel's incompleteness theorem can be performed in perfectly finitistic mathematics (reducible to the manipulation of concrete tokens, such as we perform in plain arithmetic and combinatorics of natural numbers). I don't know what specifically you object to in Godel's work. 

I am no expert, but I seem to recall the kinds of "truths" missing from a consistent system, making them "incomplete" are the sorts of things which have no meaning... self referential and/or are self-contradictory...

at the very least, and I am no expert, I have not been heard of any meaningful truth being missing from a consistent system...

Whether or not this is a real problem is also something I am not sure about... but the claims as to the ramifications, import, or profundity of the work is in many cases overblown... especially when it is used to attack knowledge as a whole, which I believe to be a gross misapplication out of context.

The reasoning is as follows:  If Gödel's system applies to systems of math... then since reality and thought are also math (Platonic Idealism) then the universe and/or man's knowledge is also complete but inconsistent or consistent but incomplete.

 

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No, it does not involve self-contradiction, and the "self-reference" (we can informally call it a kind of "self-reference" but that is to be severely qualified by understanding the exact mathematics of it) can be constructed with only statements involving generalizations about natural numbers.

Moreover, it was later proven to apply to the question of decidability of Diophantine equations, which is (putting it roughly) to answer the question whether there is an algorithm to decide for solutions to the kinds of formulas one finds in high school algebra. If one ever had the intellectual curiosity to ask "Is there an algorithm I could use to solve for unknowns in any given high school algebra problem?" then this is answered as a followup (years later) to Godel's work.

Moreover, it was later found to apply to certain other questions in mathematics that had long been a concern to mathematicians quite aside from the Godel theorem.

------

A meaningful system is Peano arithmetic. A meaningful system is PRA (which is finitisitc mathematics about natural numbers). Godel's theorem proves that there are formulas such that neither the formula nor its negation is a theorem of the system (thus that there are formulas that are true about arithmetic that cannot be derived from the axioms) and thus that there is no algorithm to decide for all formulas whether they are theorems of those systems. That is, even just onto itself, a fascinating discovery. 

-----

Possible claims about "profundity" or philosophical implications of Godel's theorem are separate from the mathematics of the Godel theorem itself. Indeed, there are lots of ridiculous claims about supposed implications of Godels' theorem; but that people misappropriate the theorem is not a fault of the mathematics of the theorem and they do not make the theorem itself "nonsense". 

-----

We don't need Godel's theorem to inform us that human knowledge is not complete. Godel (or Rosser, et. al) did not advance the work with regard to completeness of human knowledge. Rather, the work concerns specific types of formal mathematical systems. Moreover, there ARE complete and consistent formal mathematical theories; and this is well understood by anyone with a reasonable familiarity of Godels' theorem. Godel's theorem concerns only certain kinds of systems. Granted, Godel had his own philosophical views, but they do not determine the actual mathematics of the theorem.

-----

And a cousin of Godel's result is Turing's proof of the undecidability of the halting question. This implies that there is no algorithm that can determine for all computer programs whether they halt upon a given input. This has ramifications (though, I might not be able to find it again on the Internet) for the question of whether there could ever be a universal program to check for any given program whether it has a bug that causes it to not terminate (except, of course, by the physical limitation of memory in the computer). 

----

I think it is much better to learn about the actual mathematics of Godel than to opine about it based on various oversimplifications and misinformation about it on the Internet and even in books meant for popular reading rather than actual mathematical scrutiny.

Edited by GrandMinnow

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1 hour ago, GrandMinnow said:

Symbolic logic and truth tables are absolutely basic tools in the research and design of the computer you're using right now.

I should add that it is widely regarded that the most econcomically meaningful question that could be answered in mathematics now is the P v NP question. An answer would have vast consequences for the direction of the development of technologies. There is (as I recall) a one million dollar prize offered to the solution of this question. And this question takes as absolutely fundamental such things as symbolic logic and truth tables. 

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