Right? This by far has been the most efficient 6 minutes of my life in understanding the great puzzle between these philosophies. He's [I don't know is name] a great speaker
I prefer to believe that real mathematical objects do not exist in some intangible platonic dimension. Consider an invented object such as a house. We are pretty good at immediately identifying what is a ‘house’, in the same way we can identify approximations of ‘triangles’, ‘circles’, or ‘lines’. Given that we abstract such an invented object, in a way similar to how we abstract objects of mathematics, there is less reason to assume that such mathematical objects correspond to their platonic forms existing in some intangible world. Instead, our mathematical abstractions are inspired by their ubiquitous approximates, which exist in the physical world, and which arose from deterministic processes. I think it is more likely, therefore, that the human mind has evolved to (subconsciously) invent abstract models. All this isn't to say that, ultimately, truth is not abstract in nature, it's just to reject the Theory of Forms. It is inevitable that at the core of reality, there are few non-physical, abstract, and essential (perhaps pure math) principles that can give rise to physics, and broad mathematical structure is the artifact that is byproduct of those principles. Circles are not non-reducible to such principles; ie we can build circles from concept like points, direction, and space/motion.
Discovery is not contradictory to invention. The event that someone "Invented" the house, can also be described as "Someone discovered that it is possible to assemble materials in a formation which can be today recognized as a house", an invention is in a sense a discovery. This is not a validation that the concept of a house does not exist before we "invent" it, but rather it is an indication that it existed as a concept prior to the evaluation and the manifestations that we can observe. The fact that YOU could have "invented" the house or I could have "invented" it is an indication that the concept exists externally to the human psyche
@@masterbonzala yea but i think we would have to adopt a sort of skepticism about whether we can actually knpw if abstract mathematical entites can exist mind independently. Since we have no way of coming into contact with these entities except either through concrete objects (i.e. circles, cubes, etc.) Or just mental representations
I don't understand. If you define platonism and formalism in this manner, then platonism follows from formalism. Look: if we are formalists, we say, that mathematics arise from set of rules, but then that set of rules defines a platonic world created by that rules, and then we're platonists. Either their explanation of what is platonism and what is formalism are wrong, or I don't get it.
I would assume that his definition of platonism is slightly wrong, he seems to use Plato's cave theorem to talk about abstract objects but he then assumes that platonist mathematicians believe this so-called abstract plane is a set of rules also. This isn't true, platonism merely involves numbers existing nonspatiotemporally not that they hold some higher being of rules.
@luck3949 That's a good point, and I think fwebb1710 had a good reply. To add, as I understand it, formalism holds mathematics is not `about' anything at all, as quantity (`number-ness') is a secondary concern to the strict rules of mathematics. It's more ontologically agnostic.
That guy genuinely does not seem to understand Goedel's theorems. All they tell us is that any axiomatic system capable of doing arithmetic will either be complete or consistent, but never both. It says absolutely nothing about the existence of some literal trans-dimensional realm where abstract objects reside, and it is completely absurd how this guy found a way to connect those two ideas together out of nothing. This sort of garbage reaosning is exactly why philosophy has such a bad reputation.
Well if any axiomatic system can't be both complete and consistent it leaves the platonist with either accepting that their is no formal system that exists outside of our physical world or that the world is inherently contradictory which seems to mean that formalism is true too. I think he was only discussing Godels incompleteness theorem as a possible argument against Platonism.
I always thought it was pretty sketchy that platonists don’t think a shoddy, rough approximation of a circle is just as deserving of being in the “platonic realm” as a “perfect” circle. Also, i feel like godel incompleteness in the context of formalism is “you can always invent new games to play” meanwhile for platonism, how can the platonic realm even exist in a meaningful sense if it would have to be full of inconsistencies
I wish he touched on how math is so unreasonably effective. How is that so if it is all in the mind? Also, what about times when math is discovered/invented with no application in mind, then, 200 years later it is used for a great purpose. To say that math is imperical in nature or naturally in our minds seems ludicrious as it is SO COMPLEX and EFFECTIVE. On the other hand though, the seemingly arbitrary nature of choosing the axioms makes it seem so unlikely thay we have found skme ulitmate truth.
How can they end it there! I need to know the truth otherwise I'll never be able to accept what I'm learning in the classroom and therefore I'll be dragging my feet the entire way through classes wondering if I should even bother learning something that isn't even true!
I hope I'm not too late. But even if you learn something that is false or thath you probably don't have any direct practical need of knowing it may still help you with the teqnuiqes and practises you go through.
Any line of reasoning that doesn’t use or-introduction or or-elimination is kinda harmless. You can draw a lot of valid conclusions even from inconsistent foundations as we know from the study of paraconsistent logic. Given that we haven’t found any inconsistencies in ZFC so far, chances are that if there are any at all, they ought to be relatively small and this fixable. Additionally, a lot of math doesn’t even make use of the axioms that enable Gödel’s incompleteness theorem. Whenever you do finitary math, there’s no problem at all. What Gödel is actually telling us is that you can always find propositions that are logically independent of the current axiom system. That is to say: when we do sufficiently complex math, then it is a feature of correct (i.e. consistent) axiomatic systems (on which you can confidently draw conclusions) that the axiomatic system is incomplete.
There are tons of math structures that "exist" that are of no use to "humans" because they do not describe the world we live in. That is why we use the ones we do. They will however be useful in other worlds. Maths, the universal language because the universe is a mathematical object/structure.
There are uncountably many mathematical structures, but math can only be useful to describe countably many mathematical structures. So math can be useful for description of 0% of mathematical structures. So argument "math describes X well because X is mathematical structure" is invalid.
+Simple1DEA As the speaker says, the paradox is very similar to that of the village barber. Consider a set S defined as the set of all sets which are not sets of themselves. By its definition, this should mean that S is not a set of S, but then if S is not a set of itself, then surely by the definition it should be a set of itself. Herein lies the paradox known as Russell's paradox.
@@dogsdomain8458 There would not be any point, since if there were true contradictions you could prove every statement and its negation, thus making the system useless.
@@thedude882 The principle of explosion (from a contradiction, all statements are provable) is itself provable in classical logic, but as Dog's Domain mentions, there is a notion of paraconsistent logic. A paraconsistent logic is a form of logic where the the principle of explosion does not hold - where some contradiction does not imply every statement. Using some versions of paraconsistent logic, the Russell's set can exist. Newton C. A. da Costa has developed nontrivial paraconsistent versions of set theory in which Russell's set exists (provided that things like ZF are consistent).
No doubt There is a Link between réality and Mathematics as Galileo showed it and Pytagorus and Plato Both thought. B. Russel came to the same conclusion With his funny paradoxe of The barber to which our scientist référ. So man and his World are of the same Stuff. (Shakespeare)
Dont be silly, Mathematics doesnt explain the whole universe. It only explains the theoretical constructs of science, which are constructed with Mathematics, duh 😂
The only mathematical construct or mathematical object in the universe, IS the universe. The universe is a mathematical construct. A mathematical object. That is what makes it understandable to begin with and why mathematics is thee universal language.
@@sleepycorn349 The best way I can put it is that logic only uses logical language when trying to restate mathematics, while formalism will use whatever linguistic symbols it can find to express mathematics. They are similar but not the same.
In other words, Formalists would say, "Math is just a game of symbols, we don't care whether it is the truth or not". While Logicians would say, "If math is just your game, the fact it requires logic to justify it, only a set of rules would still collapse if it breaks logic (such as Russell's Paradox)"
"Mathematical Platonism" isn't truly Platonism it merely resembles it, Plato believed that an ideal world existed in reality and that this world we find ourselves in is an imperfect reflection of reality. In Plato's "Real Ideal world" exist an ideal for every true quality Truth, Solid, Wet, Beauty, etc etc. Mathematical Platonism claims that mathematics although abstract have a real and independent existence outside of being mere concepts of intelligent minds, thus math is "discovered" and not "invented.
As for meta-mathematical objects, circles/triangles being part of some Platonic realm, then yes, Plato was wrong. It's quite possible, however, if not likely, that very elementary math or logical laws are a the core of reality. Ironic that Plato claims to be so removed from the physical, as Circles are way to 'touchy'/complex. Circles can be modeled from more essential elements like points, space, motion, and spin.
Epic summation of 2000 years of thought.
Right? This by far has been the most efficient 6 minutes of my life in understanding the great puzzle between these philosophies. He's [I don't know is name] a great speaker
this talk saved my ass in the TOK essay #ibstudentsunite
+Alisa C i'm here just for that :D
Are they giving you a topic that you have to write about these days? Or did you both just happen to chose the same topic?
I hope it saves my ass now!!
I prefer to believe that real mathematical objects do not exist in some intangible platonic dimension. Consider an invented object such as a house. We are pretty good at immediately identifying what is a ‘house’, in the same way we can identify approximations of ‘triangles’, ‘circles’, or ‘lines’. Given that we abstract such an invented object, in a way similar to how we abstract objects of mathematics, there is less reason to assume that such mathematical objects correspond to their platonic forms existing in some intangible world. Instead, our mathematical abstractions are inspired by their ubiquitous approximates, which exist in the physical world, and which arose from deterministic processes. I think it is more likely, therefore, that the human mind has evolved to (subconsciously) invent abstract models. All this isn't to say that, ultimately, truth is not abstract in nature, it's just to reject the Theory of Forms. It is inevitable that at the core of reality, there are few non-physical, abstract, and essential (perhaps pure math) principles that can give rise to physics, and broad mathematical structure is the artifact that is byproduct of those principles. Circles are not non-reducible to such principles; ie we can build circles from concept like points, direction, and space/motion.
Absolutely spot on
Discovery is not contradictory to invention. The event that someone "Invented" the house, can also be described as "Someone discovered that it is possible to assemble materials in a formation which can be today recognized as a house", an invention is in a sense a discovery. This is not a validation that the concept of a house does not exist before we "invent" it, but rather it is an indication that it existed as a concept prior to the evaluation and the manifestations that we can observe. The fact that YOU could have "invented" the house or I could have "invented" it is an indication that the concept exists externally to the human psyche
@@masterbonzala Well put.
That sounds a lot like conceptualism
@@masterbonzala yea but i think we would have to adopt a sort of skepticism about whether we can actually knpw if abstract mathematical entites can exist mind independently. Since we have no way of coming into contact with these entities except either through concrete objects (i.e. circles, cubes, etc.) Or just mental representations
I don't understand. If you define platonism and formalism in this manner, then platonism follows from formalism. Look: if we are formalists, we say, that mathematics arise from set of rules, but then that set of rules defines a platonic world created by that rules, and then we're platonists. Either their explanation of what is platonism and what is formalism are wrong, or I don't get it.
I would assume that his definition of platonism is slightly wrong, he seems to use Plato's cave theorem to talk about abstract objects but he then assumes that platonist mathematicians believe this so-called abstract plane is a set of rules also. This isn't true, platonism merely involves numbers existing nonspatiotemporally not that they hold some higher being of rules.
@@fwebb1710how could numbers exist nonspatiotemporally but not hold a higher set of rules? Wouldn't that simply be that they exist in our mind?
@luck3949 That's a good point, and I think fwebb1710 had a good reply. To add, as I understand it, formalism holds mathematics is not `about' anything at all, as quantity (`number-ness') is a secondary concern to the strict rules of mathematics. It's more ontologically agnostic.
the title should be platonism versus formalism versus logicism
didn't notice Minsky until I watched it again.
That guy genuinely does not seem to understand Goedel's theorems. All they tell us is that any axiomatic system capable of doing arithmetic will either be complete or consistent, but never both. It says absolutely nothing about the existence of some literal trans-dimensional realm where abstract objects reside, and it is completely absurd how this guy found a way to connect those two ideas together out of nothing. This sort of garbage reaosning is exactly why philosophy has such a bad reputation.
Well if any axiomatic system can't be both complete and consistent it leaves the platonist with either accepting that their is no formal system that exists outside of our physical world or that the world is inherently contradictory which seems to mean that formalism is true too. I think he was only discussing Godels incompleteness theorem as a possible argument against Platonism.
I always thought it was pretty sketchy that platonists don’t think a shoddy, rough approximation of a circle is just as deserving of being in the “platonic realm” as a “perfect” circle. Also, i feel like godel incompleteness in the context of formalism is “you can always invent new games to play” meanwhile for platonism, how can the platonic realm even exist in a meaningful sense if it would have to be full of inconsistencies
I nearly cried , while I have trouble in understanding the Formalist view of mathematics.
I wish he touched on how math is so unreasonably effective. How is that so if it is all in the mind? Also, what about times when math is discovered/invented with no application in mind, then, 200 years later it is used for a great purpose. To say that math is imperical in nature or naturally in our minds seems ludicrious as it is SO COMPLEX and EFFECTIVE. On the other hand though, the seemingly arbitrary nature of choosing the axioms makes it seem so unlikely thay we have found skme ulitmate truth.
How can they end it there! I need to know the truth otherwise I'll never be able to accept what I'm learning in the classroom and therefore I'll be dragging my feet the entire way through classes wondering if I should even bother learning something that isn't even true!
I hope I'm not too late. But even if you learn something that is false or thath you probably don't have any direct practical need of knowing it may still help you with the teqnuiqes and practises you go through.
Any line of reasoning that doesn’t use or-introduction or or-elimination is kinda harmless. You can draw a lot of valid conclusions even from inconsistent foundations as we know from the study of paraconsistent logic.
Given that we haven’t found any inconsistencies in ZFC so far, chances are that if there are any at all, they ought to be relatively small and this fixable.
Additionally, a lot of math doesn’t even make use of the axioms that enable Gödel’s incompleteness theorem. Whenever you do finitary math, there’s no problem at all.
What Gödel is actually telling us is that you can always find propositions that are logically independent of the current axiom system. That is to say: when we do sufficiently complex math, then it is a feature of correct (i.e. consistent) axiomatic systems (on which you can confidently draw conclusions) that the axiomatic system is incomplete.
The Universal Law.
There are tons of math structures that "exist" that are of no use to "humans" because they do not describe the world we live in. That is why we use the ones we do. They will however be useful in other worlds. Maths, the universal language because the universe is a mathematical object/structure.
There are uncountably many mathematical structures, but math can only be useful to describe countably many mathematical structures. So math can be useful for description of 0% of mathematical structures. So argument "math describes X well because X is mathematical structure" is invalid.
What paradox in mathematics was he referring to?
+Simple1DEA As the speaker says, the paradox is very similar to that of the village barber. Consider a set S defined as the set of all sets which are not sets of themselves. By its definition, this should mean that S is not a set of S, but then if S is not a set of itself, then surely by the definition it should be a set of itself. Herein lies the paradox known as Russell's paradox.
Russel Paradox. It attacks the concept of Naive Set Theory by Georg Cantor.
@@tonyalberty7232 what if you adopt some form of paraconsistent logic and just say that in set theory, there can exist some true contradictions?
@@dogsdomain8458 There would not be any point, since if there were true contradictions you could prove every statement and its negation, thus making the system useless.
@@thedude882 The principle of explosion (from a contradiction, all statements are provable) is itself provable in classical logic, but as Dog's Domain mentions, there is a notion of paraconsistent logic. A paraconsistent logic is a form of logic where the the principle of explosion does not hold - where some contradiction does not imply every statement.
Using some versions of paraconsistent logic, the Russell's set can exist. Newton C. A. da Costa has developed nontrivial paraconsistent versions of set theory in which Russell's set exists (provided that things like ZF are consistent).
if you actually understand platonism this video should bring you pain
Bro got 2k16 highlights 😂
Mathematics is a spook
Stirner and his consequences
@ Neueregel Plato was right, and God is a mathematician. Boo!!
No doubt There is a Link between réality and Mathematics as Galileo showed it and Pytagorus and Plato Both thought. B. Russel came to the same conclusion With his funny paradoxe of The barber to which our scientist référ. So man and his World are of the same Stuff. (Shakespeare)
Dont be silly, Mathematics doesnt explain the whole universe. It only explains the theoretical constructs of science, which are constructed with Mathematics, duh 😂
The only mathematical construct or mathematical object in the universe, IS the universe. The universe is a mathematical construct. A mathematical object. That is what makes it understandable to begin with and why mathematics is thee universal language.
Formalism is logic
no they are different.
@@kamalaharriswalz2025 in what way sir? Im not a mathematician tho im just curious hehe
@@sleepycorn349 The best way I can put it is that logic only uses logical language when trying to restate mathematics, while formalism will use whatever linguistic symbols it can find to express mathematics. They are similar but not the same.
In other words, Formalists would say, "Math is just a game of symbols, we don't care whether it is the truth or not". While Logicians would say, "If math is just your game, the fact it requires logic to justify it, only a set of rules would still collapse if it breaks logic (such as Russell's Paradox)"
Plato was wrong
You are still in Platos cave . He advised to try and come out ..
ST DOMESTIHUS There is no cave and Plato is dead for 23 centuries !! ha ha
"Mathematical Platonism" isn't truly Platonism it merely resembles it, Plato believed that an ideal world existed in reality and that this world we find ourselves in is an imperfect reflection of reality. In Plato's "Real Ideal world" exist an ideal for every true quality Truth, Solid, Wet, Beauty, etc etc. Mathematical Platonism claims that mathematics although abstract have a real and independent existence outside of being mere concepts of intelligent minds, thus math is "discovered" and not "invented.
JudoMateo i agree. but many things changed since then
As for meta-mathematical objects, circles/triangles being part of some Platonic realm, then yes, Plato was wrong. It's quite possible, however, if not likely, that very elementary math or logical laws are a the core of reality. Ironic that Plato claims to be so removed from the physical, as Circles are way to 'touchy'/complex. Circles can be modeled from more essential elements like points, space, motion, and spin.