6 ms·

I used to feel the same way. I now consider complex numbers just as real as any other number. The key to seeing the light is not to try convincing yourself tha

by prmph 8mo ago

I used to feel the same way. I now consider complex numbers just as real as any other number. The key to seeing the light is not to try convincing yourself that complex number are "real", but to truly understand how ALL numbers are abstractions. This has indeed been a perspective that has broadened my understanding of math as a whole. Reflect on the fact that negative numbers, fractions, even zero, were once controversial and non-intuitive, the same as complex are to some now. Even the "natural" numbers are only abstractions: they allow us to categorize by quantity. No one ever saw "two", for example. Another thing to think about is the very nature of mathematical existence. In a certain perspective, no objects cannot exist in math. If you can think if an object with certain rules constraining it, voila, it exists, independent of whether a certain rule system prohibit its. All that matters is that we adhere to the rule system we have imagined into being. It does not exist in a certain mathematical axiomatic system, but then again axioms are by their very nature chosen. Now in that vein here is a deep thought: I think free will exists just because we can imagine a math object into being that is neither caused nor random. No need to know how it exists, the important thing is, assuming it exists, what are its properties?

anon291 8mo ago
Correct. And this is the key distinction between the mathematical approach and the everyday / business / SE approach that dominates on hacker news. Numbers are not "real", they just happen to be isomorphic to all things that are infinite in nature. That falls out from the isomorphism between countable sets and the natural numbers. You'll often hear novices referencing the 'reals' as being "real" numbers and what we measure with and such. And yet we categorically do not ever measure or observe the reals at all. Such thing is honestly silly. Where on earth is pi on my ruler? It would be impossible to pinpoint... This is a result of the isomorphism of the real numbers to cauchy sequences of rational numbers and the definition of supremum and infinum. How on earth can any person possibly identify a physical least upper bound of an infinite set? The only things we measure with are rational numbers. People use terms sloppily and get themselves confused. These structures are fundamental because they encode something to do with relationships between things The natural numbers encode things which always have something right after them. All things that satisfy this property are isomorphic to the natural numbers. Similarly complex numbers relate by rotation and things satisfying particular rotational symmetries will behave the same way as the complex numbers. Thus we use C to describe them. As a Zen Koan: A novice asks "are the complex numbers real?" The master turns right and walks away.
- meroes 8mo ago
  Very similar arguments date back to at least Plato. Ancient Greek math was based in geometry and Plato argued one could never demonstrate incommensurable lengths of rope due to physical constraints. And yet incommensurable lengths exist in math. So he said the two realms are forever divided. I think it’s modern science’s use of math that made people forget this.
  - anon291 8mo ago
    Mathematics (and computer science) is often taught independent of philosophy, which is a loss for both fields.
    - anthk 8mo ago
      Philiosophers aren't aware but Science itself and math curb-stomped most of the bullshit from philosophy and for the good. Lovecraft captured well that feeling with Cosmic Horror. But, you know, in the 20's, 30's, 40's, scifi writters evolved. Outdated, romantic foes (specially the French and German romanticism) keep bitching over and over about the pure and 'simple' past, as if the universe had a meaning per se. And they are utterly lost. Forever. Archimedes and Euclides won over Aristotle. Guess why. Math itself it's the Logos.
      - meroes 8mo ago
        I get that philosophy was outmoded in lots of ways by modern science, but Aristotle is “the father of biology”, basically starting the discipline, so the last point is hard to understand.
- NedF 8mo ago
  [dead]
danbruc 8mo ago
I think free will exists just because we can imagine a math object into being that is neither caused nor random. Can you? I can only imagine world_state(t + ε) = f(world_state(t), true_random_number_source). And even in that case we do not know if such a thing as true_random_number_source exists. The future state is either a deterministic function of the current state or it is independent of it, of which we can think as being a deterministic function of the world state and some random numbers from a true random number source. Or a mixture of the two, some things are deterministic, some things are random. But neither being deterministic nor being random qualifies as free will for me. I get the point of compatibilists, we can define free will as doing what I want, even if that is just a deterministic function of my brain state and the environment, and sure, that kind of free will we have. But that is not the kind of free will that many people imagine, being able to make different decisions in the exact same situation, i.e. make a decision, then rewind the entire universe a bit, and make the decision again. With a different outcome this time but also not being a random outcome. I can not even tell what that would mean. If the choice is not random and also does not depend on the prior state, on what does it depend? The closest thing I can imagine is your brain deterministically picking two possible meals from the menu based on your preferences and the environment respectively circumstances, and then flipping a coin to make the final decision. The outcome is deterministically constraint by your preferences but ultimately a random choice within those constraints. But is that what you think of as free will? The decision result depends on you, which option you even consider, but the final choice within those acceptable options does not depend on you in any way and you therefore have no control over it.
- prmph 8mo ago
  > But neither being deterministic nor being random qualifies as free will for me Not sure what you mean here, but non-random + non-caused is the very definition of free will. It is closely bound up with the problem of consciousness, because we need to define the "you" that has free will. It is certainly not your individual brain cells nor your organs. But irrespective of what you define "you" to be, free will is the "you"'s ability to choose, influenced by prior state but not wholly, and also not random. And, No, I am not talking about compatibilism.
Pythagomizer 8mo ago
I like this approach. I especially agree with the comparison of complex numbers to negative numbers. Remember that historically, not every civilization even had a number for zero. Likewise, mathematicians struggled with a generalized solution to the Quadratic. The problem was that there were at least 6 possible equations to solve a quadratic without using negative numbers. Back then, its application was limited to area and negative numbers seemed irrelevant based on the absolute value nature of distance. It was only by abandoning our simplistic application rooted in reality that we could develop a single Quadratic Equation and with it open a new world of possibilities.
deterministic 8mo ago
> I think free will exists just because we can imagine a math object into being that is neither caused nor random. You can absolutely be deterministic and still believe you have free will.

by prmph 8mo ago

anon291 8mo ago

Correct. And this is the key distinction between the mathematical approach and the everyday / business / SE approach that dominates on hacker news. Numbers are not "real", they just happen to be isomorphic to all things that are infinite in nature. That falls out from the isomorphism between countable sets and the natural numbers. You'll often hear novices referencing the 'reals' as being "real" numbers and what we measure with and such. And yet we categorically do not ever measure or observe the reals at all. Such thing is honestly silly. Where on earth is pi on my ruler? It would be impossible to pinpoint... This is a result of the isomorphism of the real numbers to cauchy sequences of rational numbers and the definition of supremum and infinum. How on earth can any person possibly identify a physical least upper bound of an infinite set? The only things we measure with are rational numbers. People use terms sloppily and get themselves confused. These structures are fundamental because they encode something to do with relationships between things The natural numbers encode things which always have something right after them. All things that satisfy this property are isomorphic to the natural numbers. Similarly complex numbers relate by rotation and things satisfying particular rotational symmetries will behave the same way as the complex numbers. Thus we use C to describe them. As a Zen Koan: A novice asks "are the complex numbers real?" The master turns right and walks away.

meroes 8mo ago
Very similar arguments date back to at least Plato. Ancient Greek math was based in geometry and Plato argued one could never demonstrate incommensurable lengths of rope due to physical constraints. And yet incommensurable lengths exist in math. So he said the two realms are forever divided. I think it’s modern science’s use of math that made people forget this.
- anon291 8mo ago
  Mathematics (and computer science) is often taught independent of philosophy, which is a loss for both fields.
  - anthk 8mo ago
    Philiosophers aren't aware but Science itself and math curb-stomped most of the bullshit from philosophy and for the good. Lovecraft captured well that feeling with Cosmic Horror. But, you know, in the 20's, 30's, 40's, scifi writters evolved. Outdated, romantic foes (specially the French and German romanticism) keep bitching over and over about the pure and 'simple' past, as if the universe had a meaning per se. And they are utterly lost. Forever. Archimedes and Euclides won over Aristotle. Guess why. Math itself it's the Logos.
    - meroes 8mo ago
      I get that philosophy was outmoded in lots of ways by modern science, but Aristotle is “the father of biology”, basically starting the discipline, so the last point is hard to understand.
NedF 8mo ago
[dead]

danbruc 8mo ago

I think free will exists just because we can imagine a math object into being that is neither caused nor random. Can you? I can only imagine world_state(t + ε) = f(world_state(t), true_random_number_source). And even in that case we do not know if such a thing as true_random_number_source exists. The future state is either a deterministic function of the current state or it is independent of it, of which we can think as being a deterministic function of the world state and some random numbers from a true random number source. Or a mixture of the two, some things are deterministic, some things are random. But neither being deterministic nor being random qualifies as free will for me. I get the point of compatibilists, we can define free will as doing what I want, even if that is just a deterministic function of my brain state and the environment, and sure, that kind of free will we have. But that is not the kind of free will that many people imagine, being able to make different decisions in the exact same situation, i.e. make a decision, then rewind the entire universe a bit, and make the decision again. With a different outcome this time but also not being a random outcome. I can not even tell what that would mean. If the choice is not random and also does not depend on the prior state, on what does it depend? The closest thing I can imagine is your brain deterministically picking two possible meals from the menu based on your preferences and the environment respectively circumstances, and then flipping a coin to make the final decision. The outcome is deterministically constraint by your preferences but ultimately a random choice within those constraints. But is that what you think of as free will? The decision result depends on you, which option you even consider, but the final choice within those acceptable options does not depend on you in any way and you therefore have no control over it.

prmph 8mo ago
> But neither being deterministic nor being random qualifies as free will for me Not sure what you mean here, but non-random + non-caused is the very definition of free will. It is closely bound up with the problem of consciousness, because we need to define the "you" that has free will. It is certainly not your individual brain cells nor your organs. But irrespective of what you define "you" to be, free will is the "you"'s ability to choose, influenced by prior state but not wholly, and also not random. And, No, I am not talking about compatibilism.

Pythagomizer 8mo ago

I like this approach. I especially agree with the comparison of complex numbers to negative numbers. Remember that historically, not every civilization even had a number for zero. Likewise, mathematicians struggled with a generalized solution to the Quadratic. The problem was that there were at least 6 possible equations to solve a quadratic without using negative numbers. Back then, its application was limited to area and negative numbers seemed irrelevant based on the absolute value nature of distance. It was only by abandoning our simplistic application rooted in reality that we could develop a single Quadratic Equation and with it open a new world of possibilities.

deterministic 8mo ago

> I think free will exists just because we can imagine a math object into being that is neither caused nor random. You can absolutely be deterministic and still believe you have free will.