| ▲ | tshaddox 5 days ago |
| Isn't this article conflating our formalism of a given abstract entity (like real numbers or integers) with the abstract entity itself? Surely quantities existed long before humans (e.g. there was a quantity of stars in the Milky Way 1 million years ago). And surely ordinals existed long before humans (e.g. there was a most massive star in the Milky Way 1 million years ago). The article's claim seems to be about the mathematical formalisms humans have invented for integers and real numbers. And I agree that our formalism of integers is simpler and more elegant than our formalism of real numbers. But that could just be because we've done a worse job formalizing real numbers! |
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| ▲ | EthanHeilman 5 days ago | parent | next [-] |
| > Isn't this article conflating our formalism of a given abstract entity (like real numbers or integers) with the abstract entity itself? It is arguing that the integers separate from the reals is the formalism and that the abstract entity is the reals. We also have a formalism of the reals, but it is closer to the abstract entity. > And I agree that our formalism of integers is simpler and more elegant than our formalism of real numbers. But that could just be because we've done a worse job formalizing real numbers! As we create better more useful formalisms, we interact more with the formalism than the thing itself. It is like putting on oven mitts to pick up a hot tray from the oven. This has already happened with the reals! Consider the question of if the reals are well-ordered or not. The question applies to the actual entity of the reals itself, but the absoluteness theorem shows that the question has no arithmetic consequences. You can simple ignore the question. Thus, those seeking the mental comfort and utility of the formalism do not have to concern themselves with the true nature of the real number line. |
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| ▲ | griffzhowl 4 days ago | parent [-] | | > It is arguing that the integers separate from the reals is the formalism and that the abstract entity is the reals. Why believe in abstract entities at all, as something distinct from the formalism? We have various formal or abstract concepts, that are useful in science and its applications, but they make contact with reality only through these uses: the natural numbers are used in counting, which is useful because there are many physical objects or events that are similar enough to each other that their differences can be neglected, then it's meaningful to count them and come up with a number for the collection, e.g. apples in a basket, planets in the solar system, or whatever. Real numbers developed from our perception of continuous physical magnitudes, and from the usefulness of applying the concept of number to these magnitudes. Then the formalism of the real numbers was developed based on mathematical constructions from the natural numbers/integers. We don't have to posit some abstract entity that the concept of real numbers refers to: it's a symbolic or mathematical construction that helps us in reasoning about (what we perceive as) continuous magnitudes, which aren't abstract, but concrete aspects of our experience of the world | | |
| ▲ | EthanHeilman 4 days ago | parent [-] | | > Why believe in abstract entities at all, as something distinct from the formalism? Because they seem to have some actual reality independent of human ideas. Not that everyone agrees with this, but there are good arguments for it (see the last few thousand years of debates on the subject of platonic idealism). | | |
| ▲ | griffzhowl 4 days ago | parent [-] | | I'm saying that abstract entities only seem to have an independent existence because of a misconception that words function by referring to independent entities. Some of them do, but some also get their meaning from the part they play in a procedure, like the number words in the procedure of counting, and by extension the mathematical concepts built on these. There are reasons these particualr concepts are useful, but these reasons have to do with the structure of the concrete physical world and human activities in it, not an independent platonic reality. Wittgenstein writes on this theme quite a bit. The fact that something has been discussed for thousands of years also has nothing to do with whether there are good reasons for believing it, e.g. the Earth being flat or stationary at the center of the universe. People can be wrong for thousands of years | | |
| ▲ | EthanHeilman 2 days ago | parent [-] | | > The fact that something has been discussed for thousands of years also has nothing to do with whether there are good reasons for believing it, e.g. the Earth being flat or stationary at the center of the universe. People can be wrong for thousands of years I agree people can be wrong for thousands of years, but a flat Earth was conclusively disproved early on. I'm not talking about random people debating this issue, I'm talking a continuous intellectual process over thousands of years to understand the nature of mathematics. It may turn out that various theories put forward will be shown to be wrong. This has already happened, Godel's incompleteness was a major blow to people that argued that mathematics did not have an independent reality because it was just a logical game defined by axiomatic rules. Godel showed that axiomatic systems beyond a certain level of complexity have a reality beyond that logic can investigate. | | |
| ▲ | griffzhowl 2 days ago | parent [-] | | > Godel showed that axiomatic systems beyond a certain level of complexity have a reality beyond that logic can investigate. Godel's theorems are notoriously liable to misinterpretation, and to be honest this sounds like one. What Godel actually proved is that given a consistent formal system capable of representing arithmetic, there are statements in the language of that system such that neither the statement nor its negation can be proved in the system. The thing is these statements will be different for different formal systems. The theorem doesn't say that there are statements that are in general unprovable in any formal system, which is a common misconception. Maybe that's not what you're saying, but I find it hard to relate the claims your making about Godel's theorem to the statement of the theorem itself. Godel's proof itself can be formalized, so I don't see how it places a limitation on formalism in general | | |
| ▲ | EthanHeilman 2 days ago | parent [-] | | > What Godel actually proved is that given a consistent formal system capable of representing arithmetic, there are statements in the language of that system such that neither the statement nor its negation can be proved in the system. It goes what step further, there are statements that are true but can not be proved. > Godel's theorems are notoriously liable to misinterpretation, and to be honest this sounds like one. What do you think was Godel's philosophical motivation for investigating incompleteness? "Many philosophers and logicians have explored the significance of these theorems. However, they would often find a reason to distance themselves from the interpretation their author, Kurt Gödel, ascribed them. Gödel believed that his results provide a strong argument for the objective existence of a rationally organized world of concepts, which can be to some extent described by a deductive system but cannot be changed or manipulated (cf. [12], p. 320)." [0] You can argue that Godel misinterpreted incompleteness and There is a case to be made for that. Godel probably did not misinterpret his own philosophy about incompleteness. > Godel's proof itself can be formalized, so I don't see how it places a limitation on formalism in general It uses formalism to show that formalism is limited and that the positivist notion of truth does not include things which are true but can not shown to be true (within a axiomatic system capable of expressing arithmetic). [0]: Kurt Gödel and the Logic of Concepts (2024) https://arxiv.org/html/2406.05442v2#bib.bib12 | | |
| ▲ | griffzhowl 2 days ago | parent [-] | | > It goes what step further, there are statements that are true but can not be proved. No, this is the misconception. I know it's often stated this way, but it's wrong. Godel showed that given any formal system, you can construct a statement that the system can neither prove nor disprove. But Godel actually proves this statement in his proof, so it's certainly not unprovable in general. | | |
| ▲ | EthanHeilman a day ago | parent [-] | | As far as I can tell this is not a misconception, I am just not communicating clearly enough. I figured I could make the point at a higher level. You can construct a statement in an axiomatic system complex enough to express arithmetic: 1. Which is true, 2. AND which can be proved to be true under the assumption the system is consistent, 3. and which can not be proven true within that axiomatic system because you can't prove consistency of the system itself within the axiomatic system itself. The crass positivist position would be that such undecidable statements are meaningless, i.e. neither true nor false. The incompleteness theorems provide a strong counter-example: a statement which can not be proved within that axiomatic system, but is true (if the system is consistent). The positivists must then amend their position to include things which can't be proved within the axiomatic system but can be proved true. This resulted in the watered down late-positivism. | | |
| ▲ | griffzhowl 5 hours ago | parent [-] | | Yes, the misconception is to think Godel's theorem shows that there are true but unprovable statements. In fact, what it means for a mathematical statement to be true is that it's provable. Godel showed that the concept of provability is much more subtle than following from a single axiom system, but it doesn't show that there's some kind of transcendent form of mathematical truth that doesn't depend on proof, in my view. |
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| ▲ | john-h-k 5 days ago | parent | prev | next [-] |
| > And I agree that our formalism of integers is simpler and more elegant than our formalism of real numbers. But that could just be because we've done a worse job formalizing real numbers! Everything you can express in integers you can express in reals, but there are many things expressable in reals not possible in integers. It would be surprising if the formalism for a thing that completely supersets another thing had an equally simple formalism |
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| ▲ | xscott 5 days ago | parent | next [-] | | > [...] but there are many things expressible in reals not possible in integers Are you sure there is anything we can express in the Reals that isn't an integer in disguise? The first answer might be the sqrt(2) or pi, but we can write a finite program to spit out digits of those forever (assuming a Turing machine with integer positions on a countably infinite length tape). The binary encoding of the program represents the number, and it only needs to be finite, not even an integer at infinity. Then you might say Chaitin's constant, but that's just a name for one value we don't know and can't figure out. You can approximate it to some number of digits, but that doesn't seem good enough to express it. You can prove a program can't emit all the digits indefinitely. And even if you could, is giving one Real number a name enough? Names are countable, and again arguably finite. It seems to me we can prove there are more Reals than there are Integer or Computable numbers, but we can't "express" more than a finite number of those which aren't computable. Integers in disguise. | | |
| ▲ | tshaddox 4 days ago | parent [-] | | It seems like you're suggesting that mathematicians replace the reals with the computables. This is a reasonable thing to try, and is likely of particular interest to constructivists. There's even this whole field: https://en.wikipedia.org/wiki/Computable_analysis | | |
| ▲ | xscott 4 days ago | parent [-] | | > It seems like you're suggesting that mathematicians replace the reals with the computables. No, not any more. :-) I liked the idea a year or two ago, but I've come to believe that even the Integers are too bizarre to worry about. For now, I'm content just playing with fixed and floating point, maybe with arbitrary precision. Stuff I can reason about. I just think people are a little too casual thinking they are really using the Reals. It might be like Feynman's quote about saying you understand QM. |
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| ▲ | tshaddox 5 days ago | parent | prev | next [-] | | I'm not suggesting the two formalisms should be equally simple. But surely it's not controversial to claim that formalizing the reals involves much more advanced mathematics (and runs into much deeper problems) than formalizing the integers. I'd argue that this disparity is slightly surprising, given that both the integers and the reals are ubiquitous in essentially all branches and levels of mathematics. | |
| ▲ | orlp 5 days ago | parent | prev [-] | | Actually in math it's very common for the more general system to be simpler. Compare for example the prime numbers with the integers, or general groups with finite simple groups and the monster group. |
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| ▲ | andrewla 4 days ago | parent | prev | next [-] |
| > a quantity of stars There is no fundamental unit of "star". Maybe we can talk about electrons or protons or something, but what is and is not a star is a model, not a reality. Concretely, a bundle of pre-stellar gasses at some point transitions to being a star, but when in that time spectrum does it make that transition? When in the process of stellar exhaustion does it stop being a star? |
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| ▲ | lotharcable 5 days ago | parent | prev | next [-] |
| Modern science is derived from Christian Scholasticism from the middle ages so this way of talking and thinking about science as being divinely originated is only unusual in the past couple centuries or so. It is from that era that they developed systems of rigorous debate, formal logic, and things like peered reviewed papers that we call "the scientific method". As far as the history of these sorts of mathematical discussions the concept of negative numbers didn't exist until the 15 century. I am sure that each new concept was faced with some resistance and debate on its true nature before it became widely accepted. So I am sure that somebody looking through the historical record could find all sorts of wild quotes from different theologians trying to grasp new concepts and reconcile them with existing mathmatical standards. |
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| ▲ | griffzhowl 4 days ago | parent [-] | | > Modern science is derived from Christian Scholasticism from the middle ages No, I don't think so. It seems much more based on ancient Greek geometry and logic, the Indian numeral system and Arabic algebra. Modern science really took off after Galileo, at the time when the ancient Greek works were recovered in Europe and could be synthesized with the arithmetic and algebra of these other cultures. Galileo himself credits the "divus" Archimedes as his main inspiration. What aspects of Christian scholasticism do you think developed into modern science? |
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| ▲ | crazygringo 5 days ago | parent | prev [-] |
| I mean, this is one of the deepest questions of philosophy. Do or can concepts and categories exist without the beings that create them? Does tuna casserole exist independently of humans? If not, how is the idea of the number 7 different from the idea of a tuna casserole? Or what about the concept of decision by majority, which isn't as basic as 7, but doesn't have the physicality of a casserole? |
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| ▲ | 1718627440 5 days ago | parent [-] | | > Do or can concepts and categories exist without the beings that create them No of course not, but that's not the question. The question is whether the concepts were created by the beings. In a system with 7 stars the number of stars doesn't change when all humans die or humans never even developed. | | |
| ▲ | crazygringo 4 days ago | parent [-] | | If humans never even developed, many philosophers would say the concepts of "number" and "7" and "star" would not exist. Others would say they all exist independently of humans. Some think abstract mathematical concepts are more privileged than physical categories like stars. That is the question. I can't even tell if you're arguing one side or the other because you say "of course not" followed by "doesn't change" when those appear to be contradictory positions. But of course there are a lot of subtleties here. There's no "of course" to any of it. These things take entire books to argue one side and try to refute the other. | | |
| ▲ | 1718627440 4 days ago | parent [-] | | > many philosophers If only this was the stuff humans wasted their energy on not shooting explosives to improve their ego. :-) > If humans never even developed, many philosophers would say the concepts of "number" and "7" and "star" would not exist. Yes, I'm in the camp that thinks this is just plain wrong. > abstract mathematical concepts are more privileged than physical categories like stars. I agree to that. > I can't even tell if you're arguing one side or the other Sorry, if I was unclear. You wrote: > Do or can concepts and categories exist without the beings that create them? This takes it a priori, that the categories are created by the beings. If this is true, then I think by definition this categories can't precede the beings. That's why I wrote "of course not". When things are created by beings, they don't precede them, when they are not, they do. What I already disagree with is what you already implied as a given. Honestly I don't even know how to argue for that, because this is what I think is part of the definition of the concepts and categories. When I coin some term it always happens in reaction to things I perceived with my senses. So given my senses don't lie to me, of course this is outside of me. There are also things that are in fact invented by humans, for example color names. This doesn't mean that colors don't exist independently, but the exact boundary is arbitrary. There are also concepts where I think they can be both. For example beauty. When applied to skin or shape, these are just invented, but the beauty of complexity or completeness exists outside of us. And beauty in general only exists as it points as to a thing that exists independently of us. |
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