Theists Equivocating the Empirical with the Transcendental

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Skepdick
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Skepdick »

Immanuel Can wrote: Wed Jun 22, 2022 2:12 pm That's arbitrary, really. Anything that's a "formula" is mathematical in structure, if not in referent. Inversely, pure mathematics that has no reference to reality is an imaginary exercise conducted within an artificial, closed system of symbols.
Well, it can't possibly be a closed system. Where would the interpretation/meaning of those symbols come from?

There are symbols on your screen right now. If it were a closed system you shouldn't be able to read or understand them.
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Re: Theists Equivocating the Empirical with the Transcendental

Post by godelian »

Skepdick wrote: Wed Jun 22, 2022 3:28 pm I am familiar with Curry's paradox. What I don't understand is why you think this paradox, or any paradox is a "problem".
It's just a theorem.
The following is considered a problem:
Wikipedia on "Curry's paradox" wrote: Since F is arbitrary, any logic having these rules allows one to prove everything.
Maybe read the page on the principle of explosion:
Wikipedia on "Principle of explosion" wrote: In classical logic, intuitionistic logic and similar logical systems, the principle of explosion (Latin: ex falso [sequitur] quodlibet, 'from falsehood, anything [follows]'; or ex contradictione [sequitur] quodlibet, 'from contradiction, anything [follows]'), or the principle of Pseudo-Scotus, is the law according to which any statement can be proven from a contradiction.[1] That is, once a contradiction has been asserted, any proposition (including their negations) can be inferred from it; this is known as deductive explosion.[2][3]
If every logic sentence can be proven to be true, no matter how false, then that system has a serious problem.
Skepdick wrote: Wed Jun 22, 2022 3:28 pm Nor do I understand why you think unrestricted comprehension is a "problem".
Again, allowing a contradiction in the system will lead to deductive explosion. The system will no longer have true and false logic sentences, because it will be possible to prove them all true, even if they are false.
Oxford dictionary on "axiom of comprehension" wrote: axiom of comprehension. The unrestricted axiom of comprehension in set theory states that to every condition there corresponds a set of things meeting the condition: (∃y) (y={x : Fx}). The axiom needs restriction, since Russell's paradox shows that in this form it will lead to contradiction.
Skepdick wrote: Wed Jun 22, 2022 3:28 pm You can design a consistent-but-incomplete theory about the numbers.
You can design a complete-but-inconsistent theory about the numbers.
You just can't design a consistent AND complete theory about the numbers.
That is an incorrect interpretation of Gödel's first incompleteness theorem.

Any axiomatization that contains Robinson's Q fragment of Peano Arithmetic will be inconsistent or incomplete. You cannot choose between inconsistent and incomplete by designing the system differently. Either it contains Q or else it doesn't. Furthermore, you cannot prove that a system that contains Q is consistent, because that would be in violation of Gödel's second incompleteness theorem. At best, you can demonstrate the equiconsistency with another system, such as in Gentzen's proof.

It is not possible to prove that a system containing the Q fragment is consistent. You cannot guarantee that by design either.

I have left out your derogatory remarks. I will not reply to them. In fact, I am now tired of discussing with you because you have an obnoxious personality. Bye.
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Immanuel Can
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Immanuel Can »

Skepdick wrote: Wed Jun 22, 2022 4:06 pm
Immanuel Can wrote: Wed Jun 22, 2022 2:12 pm That's arbitrary, really. Anything that's a "formula" is mathematical in structure, if not in referent. Inversely, pure mathematics that has no reference to reality is an imaginary exercise conducted within an artificial, closed system of symbols.
Well, it can't possibly be a closed system.
A "closed" system means one that is only self-referential, and is not "open" to being altered by variables from the external world. Maths is such a system, in that nothing from the external world can make 2+2 equal anything but 4, and pi is always going to be an infinite value starting with 3.14, even if it's Wednesday or a leap year. :wink:
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Skepdick »

godelian wrote: Wed Jun 22, 2022 4:09 pm The following is considered a problem:
Wikipedia on "Curry's paradox" wrote: Since F is arbitrary, any logic having these rules allows one to prove everything.
Yes, F is arbitrary and any logic having these rules allows one to prove everything.

This is a fact about the formal system. It tells you in no uncertain terms what such a system can and cannot prove.

What I don't understand is why you think that's a "problem"
godelian wrote: Wed Jun 22, 2022 4:09 pm Maybe read the page on the principle of explosion
What gives you the idea that I am unfamiliar with this principle?
godelian wrote: Wed Jun 22, 2022 4:09 pm If every logic sentence can be proven to be true, no matter how false, then that system has a serious problem.
Uh. Why are you conflating truth and provability? If every logic sentence can be proven then this system is complete with respect to provability.

Why is that a "problem" in your view?
godelian wrote: Wed Jun 22, 2022 4:09 pm Again, allowing a contradiction in the system will lead to deductive explosion.
That's not true. The principle of explosion doesn't hold in para-consistent systems.

Principle of explosion is just another theorem. It holds in some models and not in others.

If you want it to hold - chose the axiom of non-contradiction and use a consistent logic.
If you don't want it to hold - don't chose the axiom of non-contradiction and use a para-consistent logic.

Design choice.
godelian wrote: Wed Jun 22, 2022 4:09 pm The system will no longer have true and false logic sentences, because it will be possible to prove them all true, even if they are false.
I think you really need to elaborate here. Why are you conflating truth and provability? True in the system is Boolean truth, not semantic truth.

Why do you even call yourself a godelian if you don't understand that a system can't prove its own truthfulness?
godelian wrote: Wed Jun 22, 2022 4:09 pm That is an incorrect interpretation of Gödel's first incompleteness theorem.

Any axiomatization that contains Robinson's Q fragment of Peano Arithmetic will be inconsistent OR incomplete. You cannot choose between inconsistent and incomplete by designing the system differently. Either it contains Q or else it doesn't.
My interpretation is "incorrect"?!? You don't even understand what OR means. The direct implication of rejecting excluded middle is precisely THAT
some axiomatizations will be consistent; and some will be complete.
godelian wrote: Wed Jun 22, 2022 4:09 pm Furthermore, you cannot prove that a system that contains Q is consistent, because that would be in violation of Gödel's second incompleteness theorem. At best, you can demonstrate the equiconsistency with another system, such as in Gentzen's proof.
Of course I can prove that it's consistent. The consistency-proof IS what renders the system inconsistent. Absent the consistency-proof you can't say that the system is inconsistent!
godelian wrote: Wed Jun 22, 2022 4:09 pm It is not possible to prove that a system containing the Q fragment is consistent. You cannot guarantee that by design either.
It's not possible to prove it WITHIN the system. But I can always pick additional axioms.

I can take any system (F) and append the not(Consistent(F)) axiom. That ALWAYS has a model.
godelian wrote: Wed Jun 22, 2022 4:09 pm I have left out your derogatory remarks. I will not reply to them. In fact, I am now tired of discussing with you because you have an obnoxious personality. Bye.
Well if asking you questions you refuse to answer is "obnoxious" then I am "obnoxious".

No worries. I know the answer already, so no further discussion is necessary.

The axiom of non-contradiction is your religion.
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Skepdick »

Immanuel Can wrote: Wed Jun 22, 2022 4:24 pm A "closed" system means one that is only self-referential, and is not "open" to being altered by variables from the external world.
This sure sounds like an epic anthromorphism - the usual mind-projection fallacy. Symbols don't refer to anything. Humans interpreting symbols determine the referents of symbols.

Example: This sentence refers to itself.

As a closed system the symbols don't refer to anything.
As an open system the interpreter interpreting the symbols determines that the symbols are refering to themselves.

And if you name one of your pets "itself" then just as easily you can interpret the sentence as refering to your pet.
Immanuel Can wrote: Wed Jun 22, 2022 4:24 pm Maths is such a system, in that nothing from the external world can make 2+2 equal anything but 4 , and pi is always going to be an infinite value starting with 3.14, even if it's Wednesday or a leap year. :wink:
Ohhh, I get it now you are as confused as the rest of them Mathematicians.

2+2 is not always 4.

Take two glasses. Fill them up with water and heat (or cool) the water to 2 degrees celsius.
Add the two glasses of water into a new container - measure the temperature and convince yourself that 2 + 2 = 2.

And the binary expansion of pi starts with 11⋅0010010000111111011…
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Immanuel Can »

Skepdick wrote: Wed Jun 22, 2022 4:50 pm
Immanuel Can wrote: Wed Jun 22, 2022 4:24 pm A "closed" system means one that is only self-referential, and is not "open" to being altered by variables from the external world.
This sure sounds like an epic anthromorphism
I see you're having trouble understanding.

It's not that people don't "interpret" symbols. It's that the "interpretation" comes from the world external to the mathematical operation, and doesn't change the value in question.

2 +2 continues to equal 4, no matter if you call it .. + .. or ** + ** or TT & TT. The changes in words don't change the way the mathematics actually operates. Two of anything plus two more of anything will always amount to four of that thing.

And the same is true even when there's nobody in the room to see those things at all. Two rabbits left alone in a cage will soon produce more rabbits. If a human being doesn't see the operation, it won't mean there are not more rabbits when he opens the box. There still will be.

So let's not exaggerate the potentialities of interpretation. That's a postmodern folly. We don't have to fall for it.
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Skepdick »

Immanuel Can wrote: Wed Jun 22, 2022 5:00 pm I see you're having trouble understanding.
I see that in your mental muddle you are projecting your trouble onto me.
Immanuel Can wrote: Wed Jun 22, 2022 5:00 pm It's not that people don't "interpret" symbols. It's that the "interpretation" comes from the world external to the mathematical operation, and doesn't change the value in question.

2 +2 continues to equal 4, no matter if you call it .. + .. or ** + ** or TT & TT. The changes in words don't change the way the mathematics actually operates. Two of anything plus two more of anything will always amount to four of that thing.
I am literally demonstrating to you that WHAT you are adding CHANGES the meaning of the "+" operator! This is called polymorphism.
I literally demonstrating that the value of 2+2 changes depending on things external to the symbols.

If you are adding apples 2+2 = 4.
If you are adding temperatures 2+2=2.
Immanuel Can wrote: Wed Jun 22, 2022 5:00 pm And the same is true even when there's nobody in the room to see those things at all. Two rabbits left alone in a cage will soon produce more rabbits. If a human being doesn't see the operation, it won't mean there are not more rabbits when he opens the box. There still will be.

So let's not exaggerate the potentialities of interpretation. That's a postmodern folly. We don't have to fall for it.
It's not postmodernism. It's basic fucking physics and computer science.

When are you going to stop peddling your ignorance as philosophy?
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Immanuel Can »

Skepdick wrote: Wed Jun 22, 2022 5:03 pm If you are adding apples 2+2 = 4.
If you are adding temperatures 2+2=2.
I can see you don't understand the difference between an object and a temperature.

Oh well; I tried. But I didn't have much to work with, I guess. :wink:
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Skepdick »

Immanuel Can wrote: Wed Jun 22, 2022 5:31 pm
Skepdick wrote: Wed Jun 22, 2022 5:03 pm If you are adding apples 2+2 = 4.
If you are adding temperatures 2+2=2.
I can see you don't understand the difference between an object and a temperature.

Oh well; I tried. But I didn't have much to work with, I guess. :wink:
I see you don't understand that the "+" operator can add both objects and temperatures.

But I am repeating myself. You don't even understand polymorphism
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Re: Theists Equivocating the Empirical with the Transcendental

Post by godelian »

Immanuel Can wrote: Wed Jun 22, 2022 2:12 pm
godelian wrote: Wed Jun 22, 2022 7:02 am It is important to consider, that a scientific formula about the physical universe is an expression that belongs to science and not to mathematics.
That's arbitrary, really. Anything that's a "formula" is mathematical in structure, if not in referent.
A formula about the physical universe is never provable. It can only be justified by means of an experimental test report.

Therefore, the difference is indeed not in the form of the expression. They both use the same language and symbols. The difference is in their justification. A formula about the physical universe is governed by a different epistemology than one about abstract, Platonic objects. The one is provable while the other is merely testable.

Therefore, the distinction between mathematics and science is not arbitrary.
Immanuel Can wrote: Wed Jun 22, 2022 2:12 pm Inversely, pure mathematics that has no reference to reality is an imaginary exercise conducted within an artificial, closed system of symbols. So the matter is not so tidy, I think.
According to mathematical Platonism, pure mathematics is purely abstract but not imaginary. Mathematical Platonism considers pursuing the ideal of Pure Reason not to be a weakness but a strength of mathematics.

The only input accepted in mathematics are (arbitrary) axioms as understood in Aristotelian foundationalism.
Immanuel Can wrote: Wed Jun 22, 2022 2:12 pm And historically, we know that the truth is that mathematical adjectives are derived from empirical properties...men needed symbols for counting sheep.
By rigorously axiomatizing arithmetic theory, the link with the physical universe has been severed completely.

Any untruth generated by arithmetic theorems is exclusively the result of untruth in its axiomatic foundation.

There is no untruth possible in arithmetic theory that would be the result of a lack of correspondence with the physical universe.

Furthermore, there is absolutely no claim that any theorem would correspond, and the fact that it does not correspond, would not disprove it.
Immanuel Can wrote: Wed Jun 22, 2022 2:12 pm Science and maths have a cooperative relation. And truths evident in mathematics are not going to fail to be reflected in empirical realities.
By providing a tool to science, mathematics is an input for science. However, by rejecting all its correspondentist experimental test reports as irrelevant to mathematics, mathematics rejects science completely as an input or a tool.

The cooperative relation between science and mathematics is a one-way street only, because science accepts mathematics but mathematics does not accept science.
Immanuel Can wrote: Wed Jun 22, 2022 2:12 pm For example, if mathematics says a particular operation cannot be rendered in a mathematically-accurate way, it's not as if that reality is going to exist in the scientific realm.
Mathematics is not interested in whether anything it says, corresponds or not, with the physical universe. Mathematics is only interested in correspondence with its own abstract, Platonic universe.

Again, science listens to mathematics, but mathematics does not listen to science.
Immanuel Can wrote: Wed Jun 22, 2022 2:12 pm So if mathematics shows us, as it does, that an infinite regress of prerequisites cannot exist, then it's not as if the universe can possibly still be the product of an infinitely receding chain of causes. The maths simply expose the empirical truth of that.
Causality in the physical universe is a concern for science.

It is not a concern for mathematics, simply because the physical universe is of no interest and of no importance in mathematics. Mathematics only cares about the abstract, Platonic universe.

Mathematics is completely blind.

It does not accept sensory input. It also does not accept experimental test reports about sensory input.

The empirical is not considered "true", but considered completely irrelevant in mathematics.

Science cannot give feedback to mathematics because its conclusions do not entail syntactically from a purely abstract foundation.
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Immanuel Can »

godelian wrote: Thu Jun 23, 2022 2:12 am Therefore, the distinction between mathematics and science is not arbitrary.
That's half the story. The other half is that mathematics is the language of science, when it is at its most rigorous. The best example of that would be something like theoretical physics.
According to mathematical Platonism, pure mathematics is purely abstract but not imaginary. Mathematical Platonism considers pursuing the ideal of Pure Reason not to be a weakness but a strength of mathematics.
That's again only half the story. The idea of a totally inapplicable mathematics is futile. Mathematics is vindicated not merely by its formal elegance, but also by its efficacy and accuracy in representing empirical situations.
Immanuel Can wrote: Wed Jun 22, 2022 2:12 pm And historically, we know that the truth is that mathematical adjectives are derived from empirical properties...men needed symbols for counting sheep.
By rigorously axiomatizing arithmetic theory, the link with the physical universe has been severed completely.
Mathematics was originally derived from the empirical. There were two sheep before there was the number "2". The number, remember, is only adjectival.
Any untruth generated by arithmetic theorems is exclusively the result of untruth in its axiomatic foundation.
The axiomatic foundations of mathematics are not self-evident truisms, but suppostions. For example, one simple rule underwriting mathematics is that "all symbols, once established in a particular use, must continue for the entirety of that use to refer to the same quantities and properties to which it has been assigned." That is, if you begin an equation with "3", it must remain "..." for all subsequent uses in the same context.

In other words, no equivocating your symbols.

If you do equivocate them, you immediately produce untruths.

But the axiom above, the one against equivocation, is not any product of mathematics itself, Platonic or otherwise. Rather, it's a more basic axiom, in language, upon which all the reliability of the mathematics absolutely depends.

Another example might be BODMAS or BOMDAS (whichever you prefer). There is no inherent reason, no reason necessary to the mathematics themselves, that that is the right order of operations. It's a formal agreement made by mathematicians, so as to make the mathematics consistent and predictable. But it's pre-mathematical in origin. There is no formula that proves that must be the rule. It's purely conventional.

So it turns out that there are suppositions prior to the symbols. And from where do those suppositions come, since they are not themselves products of mathematical work? They come from the fact that we have (empirically) discovered that they are necessary assumptions to getting our mathematics to perform in the ways we find useful, consistent and reliable. They came from us. They came from the empirical world.
The cooperative relation between science and mathematics is a one-way street only, because science accepts mathematics but mathematics does not accept science.
Again, not quite: it seems that the symbol system we call "maths" is dependent on prior empirical realities, like propositions framed in language, like those above.
Mathematics is not interested ...
The phrase "mathematics is not interested" is as incoherent as "rocks refuse to sing." "Mathematics" are not capable of "being interested." But empirical people, those doing mathematics, are.
Causality in the physical universe is a concern for science.
Yes, but it can be represented aptly in mathematical terms. And using these terms allows us to see what is possible and not in the empirical situation the maths describe.

Since we know that infinite regress cannot be performed in mathematics, there is zero chance that it is a correct description of how the empirical universe exists. So again, the theoretical ends up serving our knowledge of the empirical, in this case.
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Re: Theists Equivocating the Empirical with the Transcendental

Post by godelian »

Immanuel Can wrote: Thu Jun 23, 2022 2:36 am That's again only half the story. The idea of a totally inapplicable mathematics is futile. Mathematics is vindicated not merely by its formal elegance, but also by its efficacy and accuracy in representing empirical situations.
Yes, but this efficacy is a concern only for scientists, engineers, and anybody else who desires to apply it to real-world situations. It is not a concern for mathematics.
Godrey Hardy wrote: We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not.
By being divorced completely from the physical universe, mathematics is indeed meaningless. Therefore, mathematics is indeed futile, i.e. useless. Still, by successfully deriving new meaningless statements from underlying ones, the only redeeming quality of mathematics is that it is ridiculous.

Science and engineering seek to be meaningful and even seek to be useful. Mathematics does not.

The fact that mathematics has turned out to be an effective enabler for science and engineering does not detract from the fact that mathematics itself is completely abstract and strives to be meaningless and useless. Nothing expresses this better than the formalist ontology of mathematics:
Wikipedia on "mathematical formalism" wrote: In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess."[1] According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all.
Since mathematics is not "about" anything at all, it is purposely meaningless and useless.
Mathematics was originally derived from the empirical. There were two sheep before there was the number "2". The number, remember, is only adjectival.
Not "derived" but "inspired". After proper axiomatization, this source of inspiration became completely irrelevant.

For example, nonstandard natural numbers in arithmetic theory cannot be seen or used in the physical universe. It is physically not possible to inject or even detect a transfinite number in the physical universe. However, nonstandard natural numbers certainly do exist in the abstract, Platonic world of mathematics. There is, however, no hope whatsoever that they could ever correspond to anything in the physical universe. The correspondence theory of truth is therefore inapplicable to mathematics.
Another example might be BODMAS or BOMDAS (whichever you prefer). There is no inherent reason, no reason necessary to the mathematics themselves, that that is the right order of operations. It's a formal agreement made by mathematicians, so as to make the mathematics consistent and predictable. But it's pre-mathematical in origin. There is no formula that proves that must be the rule. It's purely conventional.
Operator precedence is an issue that only exists in the infix notation, which is indeed ambiguous. It does not exist in the postfix or prefix notations, because these alternative notations are not ambiguous.

Infix notation along with the Eulerian notation for function application are costly conventions, because they are so ambiguous. These things tremendously complicate the construction of compiler front ends. If you switch to prefix notation, such as in Lisp, the compiler has a much simpler core. If you switch to postfix, such as in assembler, there is not even a need for a real compiler front end.
So it turns out that there are suppositions prior to the symbols. And from where do those suppositions come, since they are not themselves products of mathematical work? They come from the fact that we have (empirically) discovered that they are necessary assumptions to getting our mathematics to perform in the ways we find useful, consistent and reliable. They came from us. They came from the empirical world.
We avoid using them wherever we can. But then again, because infix is what they teach at school, we cannot confront people with the more efficient use of prefix or postfix notation. Instead, we built complicated front ends to accommodate this otherwise inefficient convention.

We often have to stay compatible with all the miracles and all the horrors of the past. One reason why Lisp has lost the programming languages war, is because they chose to switch to prefix notation. It allows for treating source code as data; just some other nested lists to deal with, which allows for macros that are able to process them. The Lisp people had hoped that the benefits afforded by the ability to treat code as data would compensate for programmers' lack of familiarity with the prefix notation. That was a big mistake. Most programmers prefer to drop macros and just keep the infix notation instead.

Ignoring the legacy of a gigantic installed base, is almost always a mistake. Billions of people have invested massively in learning to use all kinds of conventional monsters.

For example, POSIX is a horror story. Especially libc is a nightmare full of highly inconsistent conventions, some of which are even contradictory and even outright buggy. Will we ever be able to replace these things? No, I don't think so.
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Re: Theists Equivocating the Empirical with the Transcendental

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godelian wrote: Thu Jun 23, 2022 3:39 am Yes, but this efficacy is a concern only for scientists, engineers, and anybody else who desires to apply it to real-world situations. It is not a concern for mathematics.
That's not true, unless you want to pretend computational complexity theory doesn't exist.

Go ahead and provide a decision procedure for y > 0, where y is an integer.
Now do it where y is an infinite precision real number.

There's something Mathematical to be said about the runtime efficneicy of those two algorithms, no?
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Skepdick »

godelian wrote: Thu Jun 23, 2022 3:39 am We often have to stay compatible with all the miracles and all the horrors of the past.
...
Ignoring the legacy of a gigantic installed base, is almost always a mistake. Billions of people have invested massively in learning to use all kinds of conventional monsters.
Yeah, like set theory :lol: :lol: :lol:
godelian wrote: Thu Jun 23, 2022 3:39 am One reason why Lisp has lost the programming languages war, is because they chose to switch to prefix notation. It allows for treating source code as data; just some other nested lists to deal with, which allows for macros that are able to process them. The Lisp people had hoped that the benefits afforded by the ability to treat code as data would compensate for programmers' lack of familiarity with the prefix notation. That was a big mistake. Most programmers prefer to drop macros and just keep the infix notation instead.
Treating code as data has nothing to do with infix, prefix or postfix notation. It's an entirely separate semantic property of the langauge. It'e called homoiconicity

In fact, any Assembly language is homoiconic. As is the untyped lambda calculus. But for some reason you think that's "problematic".

You know which language is not homoiconic? Set theory.
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Re: Theists Equivocating the Empirical with the Transcendental

Post by Dontaskme »

Immanuel Can wrote: Wed Jun 22, 2022 4:24 pm
A "closed" system means one that is only self-referential, and is not "open" to being altered by variables from the external world. Maths is such a system, in that nothing from the external world can make 2+2 equal anything but 4,
Nonsense..nothing can make 2+2 =4 without the projection of those symbols appearing external to the knower.

The notion of 'self-referential' is meaningless without an openess to receive a meaning. Meaning in the sense that something is known...That openess is known as the self.

The self can only refer to itself as the knower, there is no other thing that knows. So there's no such notion as....'' A "closed" system means one that is only self-referential ...''
Because...The knower is infinitely open.

But then you do like to express what is known as a whole lot of backward forwarding nonsense...IC

The symbols 2 + 2 + 4 must always be external to the self...if those symbols were not external to the self - the self wouldn't be able to project them onto it's screen of consciousness to be seen and known.

Now all we've got to figure out is...what the dickens heck is this >>>>
A "closed" system means one that is only self-referential, and is not "open" to being altered by variables from the external world. Maths is such a system, in that nothing from the external world can make 2+2 equal anything but 4,

...........supposed to mean exactly :roll: :roll: :roll: :roll:








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