I think it can be resolved by having only one actual infinity. "Aristotle handled the topic of infinity in Physics and in Metaphysics. He distinguished between actual and potential infinity. Actual infinity is completed and definite, and consists of infinitely many elements. Potential infinity is never complete: elements can be always added, but never infinitely many." - Wikipedia
My own suspicion is that there is only on actual infinity, but it is not an infinity of elements, otherwise you have multiple infinities within that infinity (e.g., infinite number of even elements and infinite number of odd elements)
Thanks so much! That means a lot! About your question, Aristotle thinks the primary things that exist are substances, things like wood, horse, water, human. However, these substances have different properties that inhere in them, such as shape, position and quantity. Quantity is the basis of all number. Everything has some amount of stuff to it, and we can divide this stuff up and compare it to other stuff. So, it's the substance that exists, and the quantity of the stuff is a real aspect of it. Numbers, however, don't exist by themselves. They are abstractions based on the quantity in substances. This doesn't mean numbers aren't real, but they aren't real in themselves. They are real insofar as the mind is abstracting them from real things: substances. One worry might be that since Aristotle also thought the universe was finite, we will only have a finite number of numbers to work with. No worries, though. We can mentally repeat the process of adding one number based on the real process of adding quantity, and we can do this an indefinite number of times. So, there are not an actually infinite number of numbers, but a potentially infinite number of numbers since we can repeat the process of number generation an indefinite number of times. Uhm...I''m not sure I explained *that* too well, so let me know!
I don't buy the idea that the actual infinite can be a logically coherent idea but cannot be metaphysically instantiated. Our understanding of mathematics is intrinsically tied to our understanding of space and time. You might say that our understanding of math is actually just a schema reprisenting our understanding the ideal properties of an ideal space. If the actual infinit makes sense, then it must be metaphysically instantiable at least in principle. But that's a big IF. Anyways, I think Aristotle is half right. I think some applications of actual infinity are coherent. Buy so far as the notion of actually infinite divisibility goes, it is incoherent. That is why we have Zeno paradoxes.
I'm totally with you on the metaphysics underpinning the math. There are some philosophers of math (not necessarily me, but tbh I'm not entirely sure where I stand on math yet) that can account for this. Inuitionists believe math is a process whereby you build numbers based on a first item and the 'successor' relation. This isn't how they do it, but let me give you a brief sketch where you can connect intuitionism to metaphysics, but not have an actual infinite. Start with any random object, like a pencil or a book. That makes the following statement true: "There is a single (one) object." Add a second object. That makes the following statement true: "There is one more object." This second statement could be considered an operation: 'plus one'. Using it, we can generate a new number: 2. What's more, we can use this operation to generate an unlimited amount of new numbers after that. This process is grounded in a metaphysics of physical objects because they make the first number and the operation true. We can use this process to generate numbers beyond the actual amount of things (e.g., if only two objects existed, we could still generate the number three using the first object and the "plus one" operation). But, along with what Aristotle says, we would never have an actual infinite number of numbers because we don't have the ability to actually perform this operation an infinite number of times. I'm not sure if that will work, but it seems plausible to me at first blush.
🤣🤣🤣 yeah, I struggled with editing! I had some students tell me to strive for no “wasted frames-“ i.e., copious jump cuts. But I realize now that style of editing just doesn’t work for video you have to think about. And it sucks to edit that much too
Well, I’m not sure that we need an actual infinity to exist or even be possible to get set theory or even modern math. All we need to do is posit certain axioms and see what logically follows. This doesn’t make an actual infinity metaphysically possible or even coherent. It just means that given the rules you set, you can have some possible outcome.
You cannot formulate set theory without actual infinity.. potential infinity as defined by Aristotle does not lend itself to be unified as a whole.. Cantor defined the first infinite as the cardinality of the set of all natural finite numbers .. such a unification is rejected by Aristotle on the ground that unification applied on infinite domain is incoherent.. basically, cantor invented set theory in order to establish the actual infinite as a well defined mathematical concept .. you may wanna check also Paris Theorem which proves the necessity of actual infinity to tackle certain mathematical problems like the convergence of Goodstein sequence .. check also Dedekind cut which uses infinity to define irrational number as two infinite sequences of rational numbers
@@bastabey2652 Thanks for those suggestions! I haven't looked into Paris Theorem and I haven't heard about Dedekind's strategy for defining irrational numbers. Sounds fascinating and I'll definitely check them out! Do you happen to have any suggestions for good explanations of that stuff for non-mathematicians? As far as Cantor's project vs. Aristotle's objection, let me explain myself a little better and see what you think. I'm not a mathematician and I am DEFINITELY open to the possibility that I don't understand this well enough! I guess what I meant to say is that a concept's use in a formal system isn't necessarily evidence it refers to a real, possible, or even coherent thing outside of that formal system and in the real world. For example, imaginary numbers are critical to math and even though they give us real world results in fields like electrical engineering, it isn't coherent to say I can have square root of negative one hot dogs. In the same way, I think Aristotle is claiming an actual infinity is incoherent because of real-world paradoxes, and not because it is unusable in a formal system. Incidentally, I think this was Hilbert's point as well, but I haven't read enough of him to say that with confidence. So, when Cantor rejects the assumption that a proper subset has less members than its superset, he is totally free to do so and the result is certainly a mathematically respectable concept, but I still think that is a real-world incoherent idea. I think it's kind of like saying purple smells like grape. This doesn't violate any logical or semantical laws, but it doesn't make sense. Again, I am TOTALLY open to the fact that I might be wrong about this! What do you think?
The question concerning the nature of reality and whether actual infinity is real or not is not related to the definition of infinity and whether we can even define infinity .. if you go through the route of confining the real to the world of the sensible , then the whole metaphysics is compromised.. moreover, rendering math as a toolbox for engineering and science limits the scope of math as a foundational philosophical rational art .. regardless whether platonic mathematical ideals are fairytales or not, a well defined coherent actual infinity is necessary for the proper functioning modern math .. Aristotle potential infinity ain’t enough .. I recommend two books: The Infinite by Moore, and The Philosophy of Set Theory by Mary Tiles .. Happy holidays
This is awesome, thanks so much! I’ve read Moore’s book and loved it. I’ll buy the other one tonight. I am going to do some videos on infinity in calculus and set theory soon. I’d love to hear your feedback!
aristoteles was wrong. infinity is not a process. infinity when you have let the limit go to infinity. Thats when you got the actual infinity. There so called paradoxes of infinity isnot a problem. There is no contradictions. Aristoteles was wrong.
Great video. Did you ever make that philmath playlist??
Wow, this was very well articulated. I've subscribed!
Mukumhang Limbu thanks so much!!!
Thanks again!
I think it can be resolved by having only one actual infinity. "Aristotle handled the topic of infinity in Physics and in Metaphysics. He distinguished between actual and potential infinity. Actual infinity is completed and definite, and consists of infinitely many elements. Potential infinity is never complete: elements can be always added, but never infinitely many." - Wikipedia
My own suspicion is that there is only on actual infinity, but it is not an infinity of elements, otherwise you have multiple infinities within that infinity (e.g., infinite number of even elements and infinite number of odd elements)
Great video ! 👍
Thanks so much!
Goddamn, thank you, Aristotle's explanation on book iv is quite hard to tackle
Glad I could be of help!
Excellent video! You've explained a difficult topic very well! Did Aristotle say anything about numbers, whether they're REAL and possibly infinite?
Thanks so much! That means a lot!
About your question, Aristotle thinks the primary things that exist are substances, things like wood, horse, water, human. However, these substances have different properties that inhere in them, such as shape, position and quantity. Quantity is the basis of all number. Everything has some amount of stuff to it, and we can divide this stuff up and compare it to other stuff. So, it's the substance that exists, and the quantity of the stuff is a real aspect of it. Numbers, however, don't exist by themselves. They are abstractions based on the quantity in substances. This doesn't mean numbers aren't real, but they aren't real in themselves. They are real insofar as the mind is abstracting them from real things: substances.
One worry might be that since Aristotle also thought the universe was finite, we will only have a finite number of numbers to work with. No worries, though. We can mentally repeat the process of adding one number based on the real process of adding quantity, and we can do this an indefinite number of times. So, there are not an actually infinite number of numbers, but a potentially infinite number of numbers since we can repeat the process of number generation an indefinite number of times. Uhm...I''m not sure I explained *that* too well, so let me know!
I’m not sure if Aquinas was right, that we can know an infinite number of things, but perhaps we can forget an infinite number of things? 😅
I don't buy the idea that the actual infinite can be a logically coherent idea but cannot be metaphysically instantiated. Our understanding of mathematics is intrinsically tied to our understanding of space and time. You might say that our understanding of math is actually just a schema reprisenting our understanding the ideal properties of an ideal space. If the actual infinit makes sense, then it must be metaphysically instantiable at least in principle. But that's a big IF.
Anyways, I think Aristotle is half right. I think some applications of actual infinity are coherent. Buy so far as the notion of actually infinite divisibility goes, it is incoherent. That is why we have Zeno paradoxes.
I'm totally with you on the metaphysics underpinning the math. There are some philosophers of math (not necessarily me, but tbh I'm not entirely sure where I stand on math yet) that can account for this. Inuitionists believe math is a process whereby you build numbers based on a first item and the 'successor' relation. This isn't how they do it, but let me give you a brief sketch where you can connect intuitionism to metaphysics, but not have an actual infinite.
Start with any random object, like a pencil or a book. That makes the following statement true: "There is a single (one) object." Add a second object. That makes the following statement true: "There is one more object." This second statement could be considered an operation: 'plus one'. Using it, we can generate a new number: 2. What's more, we can use this operation to generate an unlimited amount of new numbers after that. This process is grounded in a metaphysics of physical objects because they make the first number and the operation true. We can use this process to generate numbers beyond the actual amount of things (e.g., if only two objects existed, we could still generate the number three using the first object and the "plus one" operation). But, along with what Aristotle says, we would never have an actual infinite number of numbers because we don't have the ability to actually perform this operation an infinite number of times.
I'm not sure if that will work, but it seems plausible to me at first blush.
At the beginning of your video I could have sworn you had an actual infinity of jump cuts in your presentation. But then it ended.
🤣🤣🤣 yeah, I struggled with editing! I had some students tell me to strive for no “wasted frames-“ i.e., copious jump cuts. But I realize now that style of editing just doesn’t work for video you have to think about. And it sucks to edit that much too
Muito bom
obrigado!
No actual infinity .. no set theory ... no modern math
Well, I’m not sure that we need an actual infinity to exist or even be possible to get set theory or even modern math. All we need to do is posit certain axioms and see what logically follows. This doesn’t make an actual infinity metaphysically possible or even coherent. It just means that given the rules you set, you can have some possible outcome.
You cannot formulate set theory without actual infinity.. potential infinity as defined by Aristotle does not lend itself to be unified as a whole.. Cantor defined the first infinite as the cardinality of the set of all natural finite numbers .. such a unification is rejected by Aristotle on the ground that unification applied on infinite domain is incoherent.. basically, cantor invented set theory in order to establish the actual infinite as a well defined mathematical concept .. you may wanna check also Paris Theorem which proves the necessity of actual infinity to tackle certain mathematical problems like the convergence of Goodstein sequence .. check also Dedekind cut which uses infinity to define irrational number as two infinite sequences of rational numbers
@@bastabey2652 Thanks for those suggestions! I haven't looked into Paris Theorem and I haven't heard about Dedekind's strategy for defining irrational numbers. Sounds fascinating and I'll definitely check them out! Do you happen to have any suggestions for good explanations of that stuff for non-mathematicians?
As far as Cantor's project vs. Aristotle's objection, let me explain myself a little better and see what you think. I'm not a mathematician and I am DEFINITELY open to the possibility that I don't understand this well enough! I guess what I meant to say is that a concept's use in a formal system isn't necessarily evidence it refers to a real, possible, or even coherent thing outside of that formal system and in the real world. For example, imaginary numbers are critical to math and even though they give us real world results in fields like electrical engineering, it isn't coherent to say I can have square root of negative one hot dogs. In the same way, I think Aristotle is claiming an actual infinity is incoherent because of real-world paradoxes, and not because it is unusable in a formal system. Incidentally, I think this was Hilbert's point as well, but I haven't read enough of him to say that with confidence.
So, when Cantor rejects the assumption that a proper subset has less members than its superset, he is totally free to do so and the result is certainly a mathematically respectable concept, but I still think that is a real-world incoherent idea. I think it's kind of like saying purple smells like grape. This doesn't violate any logical or semantical laws, but it doesn't make sense.
Again, I am TOTALLY open to the fact that I might be wrong about this! What do you think?
The question concerning the nature of reality and whether actual infinity is real or not is not related to the definition of infinity and whether we can even define infinity .. if you go through the route of confining the real to the world of the sensible , then the whole metaphysics is compromised.. moreover, rendering math as a toolbox for engineering and science limits the scope of math as a foundational philosophical rational art .. regardless whether platonic mathematical ideals are fairytales or not, a well defined coherent actual infinity is necessary for the proper functioning modern math .. Aristotle potential infinity ain’t enough .. I recommend two books: The Infinite by Moore, and The Philosophy of Set Theory by Mary Tiles .. Happy holidays
This is awesome, thanks so much! I’ve read Moore’s book and loved it. I’ll buy the other one tonight. I am going to do some videos on infinity in calculus and set theory soon. I’d love to hear your feedback!
aristoteles was wrong. infinity is not a process. infinity when you have let the limit go to infinity. Thats when you got the actual infinity.
There so called paradoxes of infinity isnot a problem. There is no contradictions.
Aristoteles was wrong.
There are only nine numbers...
Do you mean the digits?