Our server costs ~$56 per month to run. Please consider donating or becoming a Patron to help keep the site running. Help us gain new members by following us on Twitter and liking our page on Facebook!
(September 30, 2018 at 10:11 pm)Kernel Sohcahtoa Wrote: IMO, regardless of one's skill level, a student of mathematics will encounter concepts and ideas that are challenging: it seems that, for the time being, there will always be problems to solve and statements to prove. Hence, for any students who are interested in studying mathematics, please don't get discouraged when you hit a bump/boulder in the road: you are not alone. Like so many mathematicians and students of mathematics before you, embrace the fact that you are being challenged, ask questions, and allow yourself to grow from this experience, regardless of whether you are successful or not.
I'll paraphrase something that I read many decades ago, "Before anything was done using a computer, you could tell a working mathematician by how many pieces of paper were in the waste basket."
That "guess and check" method you learned in grade school has its provenance in that.
If you get to thinking you’re a person of some influence, try ordering somebody else’s dog around.
October 1, 2018 at 1:54 am (This post was last modified: October 1, 2018 at 1:55 am by robvalue.)
(September 30, 2018 at 10:11 pm)Kernel Sohcahtoa Wrote: IMO, regardless of one's skill level, a student of mathematics will encounter concepts and ideas that are challenging: it seems that, for the time being, there will always be problems to solve and statements to prove. Hence, for any students who are interested in studying mathematics, please don't get discouraged when you hit a bump/boulder in the road: you are not alone. Like so many mathematicians and students of mathematics before you, embrace the fact that you are being challenged, ask questions, and allow yourself to grow from this experience, regardless of whether you are successful or not.
Absolutely. When growing up, I was used to finding maths easy. I've just been been lucky that my brain works that way. But at certain points in my education, I crashed into brick walls. Suddenly I felt like an absolute dunce. I then had to work really hard to make progress.
Feel free to send me a private message.
Please visit my website here! It's got lots of information about atheism/theism and support for new atheists.
October 1, 2018 at 8:06 am (This post was last modified: October 1, 2018 at 8:09 am by polymath257.)
(September 30, 2018 at 10:44 pm)Fireball Wrote:
(September 30, 2018 at 10:11 pm)Kernel Sohcahtoa Wrote: IMO, regardless of one's skill level, a student of mathematics will encounter concepts and ideas that are challenging: it seems that, for the time being, there will always be problems to solve and statements to prove. Hence, for any students who are interested in studying mathematics, please don't get discouraged when you hit a bump/boulder in the road: you are not alone. Like so many mathematicians and students of mathematics before you, embrace the fact that you are being challenged, ask questions, and allow yourself to grow from this experience, regardless of whether you are successful or not.
I'll paraphrase something that I read many decades ago, "Before anything was done using a computer, you could tell a working mathematician by how many pieces of paper were in the waste basket."
That "guess and check" method you learned in grade school has its provenance in that.
I just want to say that this is still very much true today. Computers help some with computations. They are good for writing up results. But the *real* ideas happen on paper or the black (marker) board. Reams of paper are used before the first key on the computer is hit.
(September 30, 2018 at 10:11 pm)Kernel Sohcahtoa Wrote: IMO, regardless of one's skill level, a student of mathematics will encounter concepts and ideas that are challenging: it seems that, for the time being, there will always be problems to solve and statements to prove. Hence, for any students who are interested in studying mathematics, please don't get discouraged when you hit a bump/boulder in the road: you are not alone. Like so many mathematicians and students of mathematics before you, embrace the fact that you are being challenged, ask questions, and allow yourself to grow from this experience, regardless of whether you are successful or not.
Absolutely. If you are studying math to any level you *will* come across ideas that are hard to wrap your mind around. When that happens, read other books, ask people who might know, play with a hundred examples. Do whatever it takes because it will be *worth* it. Pushing through the hard stuff is much more rewarding than skimming through the easy stuff.
Eh, I guess I have something I can contribute here.
Axiomatization
Axioms in mathematics are the basic foundations of entire bodies of mathematics. Though originally the concept of an axiom in mathematics was the same as in philosophy (a basic statement so simple and obvious that everyone will agree with it), its meaning has since morphed into something subtly different. Nowadays, axioms are the minimum requirements of any well-established body of mathematics (such as arithmetic, geometry, group theory, etc). Pretty much every theorem in that body of mathematics can be derived from those basic axioms (though sometimes we'll be in subsections of those bodies that require additional axioms).
There's some philosophical debate about what axioms actually ARE. In my opinion, they're the gatekeepers / minimum conditions of that body of math, the "if" of an if-then conditional assertion. If you are dealing with an application where the axioms hold, then all the theorems of that body of math can be used with deductive certainty. However, if you're dealing with an application where some of the axioms aren't true, then all bets are off. In a sense, the tools of math are only designed and guaranteed only for situations where their axioms hold, and using them elsewhere is a bad idea. This makes axioms a bit like a checklist that you run through before using that body of mathematics.
For example, one of the axioms of basic natural number arithmetic (kinda, I might qualify this in another post) is that if you keep incrementing a number (adding 1 to it), you will never loop around back to somewhere that you've been before, but will instead proceed through an infinity of new numbers. For example, if you start counting at 4 and go up through 5, 6, 7, and so on, no matter how long you go, you will never get back to 4. Now this might seem intuitively true in the sense of a philosophical axiom, but it's not that hard to find a situation where this axiom DOESN'T hold. Imagine adding (physically combining) one pile of sand to another pile of sand. You end up with a bigger pile of sand, but it's still ONE pile of sand. Here, 1 + 1 = 1. Does this mean arithmetic's false? Not at all. Instead, it violates the don't-loop axiom, and so it falls outside of the realm of arithmetic. It's akin to wanting to loosen a Phillip's-head screw, and reaching for a ball-peen hammer. You've chosen the wrong tool for the job... and it's your fault, not the tool's.
The first and arguably most-famous axiomatization was Euclid's treatment of geometry from sometime around 300 BCE, which he founded on five postulates. (Meaning axioms. Different name for the same thing.) This was back when an axiom was the same thing in math and philosophy, and Euclid was seriously putting them forward as axioms in the philosophical sense. He formulated all of geometry (as known in his day) as a series of logical deductions derived from these five postulates. In an example of intellectual integrity for the ages, Euclid openly admitted in Elements that he had a lot of doubt about the fifth postulate. He worked very hard to prove everything he could using just the first four postulates, and the stuff that depended on the fifth postulate he flagged as perhaps being unreliable. (The controversy around the fifth postulate is a mathematical epic spanning over two millenia that I won't try to summarize in this post.) Most of the things that Euclid published in Elements were already known, but Euclid's approached the material with a level of mathematical rigor that was nearly unheard-of. Plato's Academy would sport a sign reading "let none ignorant of geometry enter", because geometry after Euclid was seen as the benchmark of logical reasoning, while other areas of math such as arithmetic had not benefited from that rigorous treatment. (In time Greek philosophy would warp itself around geometry. Geometry came to be seen as noble-math and other fields as lesser. Concepts that could not be expressed in Euclidean geometry, such as nullity, negative numbers, the infinite, and the infinitesimal became derided and dismissed as impossibilities. Social philosophies came to be based on geometry, such as nobility being justified by comparison of large similar triangles to smaller ones.)
Let's jump ahead to Renaissance and modern times. Old doctrines were being questioned, including mathematical doctrines. 0 had been imported to Europe with Arabic symbols. Algebra was also imported from the Arab world by way of Spain. Descartes created Analytic Geometry, which essentially took most or all of geometry and boiled it down to algebra -- turning noble math into infidel peasant math. People started playing around with the infinite and the infinitesimal... and Leibniz and/or Newton invented calculus.
Calculus was in particular need of mathematical rigor because no one, not even its inventors, really understood HOW it worked. Newton thought of it in terms of "fluxions" (and the original name was "The Calculus of Fluxions"), where fluxions were variables that would become zero after some mathematical simplification was performed. The problem with this was that it was a cheat. Everyone knew that his technique had no basis in rigorous mathematics. One of his detractors would complain that he was making pacts with the ghosts of numbers departed. Yet at the same time, everyone could tell that his math WORKED.
Around the 19th century, with Calculus becoming increasingly important, other bodies of mathematics emerging, and geometries that called the Euclidean model into serious doubt, a big push was made to reestablish mathematical rigor. Most of our axiomatizations date back to either the 19th or early 20th centuries. The thing is, multiple mathematicians were working on this simultaneously, and they produced multiple different axiomatizations for most of the bodies of mathematics. I've seen about a half-dozen different ways to axiomatize real numbers, to name just one example. Which brings us to the last point I want to make about sets of axioms. In some situations these different axiomiazations lead to subtle differences and we have to be clear which we're working with. (For example, ZF versus ZFC in set theory. The difference between the two, the axiom of choice, is a bright candy shell of "oh obviously that has to be true" covering a sticky caramel center of "WTAF?!?!?!?") In other cases, different axiomatizations produce exactly the same results, even with different axioms. We call these axiomatizations equivalent, and we can prove that Axiomatization A = Axiomatization B by showing that every Axiom of A is a theorem of B and vice-versa. As a result, we can't really say what the axioms of any given body of math are... because most bodies of math have multiple, equivalent formulations of their axioms, and we can't really say that one is any more right than the other because they all amount to the same thing.
BTW, the 19th century's push for rigor ended spectacularly (by mathematician standards of spectacle) with Russel's Paradox. Or as clickbait would describe it, "How Bertrand Russel DESTROYED MATHEMATICS with ONE SIMPLE QUESTION!" But that's a story for another time.
Being an antipistevist is like being an antipastovist, only with epistemic responsibility instead of bruschetta.
(October 1, 2018 at 8:06 am)polymath257 Wrote: I just want to say that this is still very much true today. Computers help some with computations. They are good for writing up results. But the *real* ideas happen on paper or the black (marker) board. Reams of paper are used before the first key on the computer is hit.
Indeed. Computers with some software are nice mostly for presentation and whatnot. It is too time consuming and dostracting havint to hotkey the math symbols in, makes your train of though less efficient.
Good ol paper or a chalk/marker board. I made my own, my ex boss has a factory that uses nice white acrylic and has tools, so I asked him if I could do one on a saturday. 1.5 * 1 meter with the support to keep it standing. He offered me the materials. It was nice of him. Still have it, but I could replace the whiteboard, Acrylic tends to get yellowish over time.
October 2, 2018 at 7:58 am (This post was last modified: October 2, 2018 at 8:10 am by polymath257.)
(October 2, 2018 at 7:40 am)LastPoet Wrote:
(October 1, 2018 at 8:06 am)polymath257 Wrote: I just want to say that this is still very much true today. Computers help some with computations. They are good for writing up results. But the *real* ideas happen on paper or the black (marker) board. Reams of paper are used before the first key on the computer is hit.
Indeed. Computers with some software are nice mostly for presentation and whatnot. It is too time consuming and dostracting havint to hotkey the math symbols in, makes your train of though less efficient.
Good ol paper or a chalk/marker board. I made my own, my ex boss has a factory that uses nice white acrylic and has tools, so I asked him if I could do one on a saturday. 1.5 * 1 meter with the support to keep it standing. He offered me the materials. It was nice of him. Still have it, but I could replace the whiteboard, Acrylic tends to get yellowish over time.
There is a specialist typesetting language called TeX (or LaTeX) that is now standard for mathematics papers. No hotkeys, but there is a learning curve for how to use it. MiKTeX is the most often used Windoze port.
(October 2, 2018 at 2:51 am)Reltzik Wrote: Eh, I guess I have something I can contribute here.
Axiomatization
Axioms in mathematics are the basic foundations of entire bodies of mathematics. Though originally the concept of an axiom in mathematics was the same as in philosophy (a basic statement so simple and obvious that everyone will agree with it), its meaning has since morphed into something subtly different. Nowadays, axioms are the minimum requirements of any well-established body of mathematics (such as arithmetic, geometry, group theory, etc). Pretty much every theorem in that body of mathematics can be derived from those basic axioms (though sometimes we'll be in subsections of those bodies that require additional axioms).
There's some philosophical debate about what axioms actually ARE. In my opinion, they're the gatekeepers / minimum conditions of that body of math, the "if" of an if-then conditional assertion. If you are dealing with an application where the axioms hold, then all the theorems of that body of math can be used with deductive certainty. However, if you're dealing with an application where some of the axioms aren't true, then all bets are off. In a sense, the tools of math are only designed and guaranteed only for situations where their axioms hold, and using them elsewhere is a bad idea. This makes axioms a bit like a checklist that you run through before using that body of mathematics.
For example, one of the axioms of basic natural number arithmetic (kinda, I might qualify this in another post) is that if you keep incrementing a number (adding 1 to it), you will never loop around back to somewhere that you've been before, but will instead proceed through an infinity of new numbers. For example, if you start counting at 4 and go up through 5, 6, 7, and so on, no matter how long you go, you will never get back to 4. Now this might seem intuitively true in the sense of a philosophical axiom, but it's not that hard to find a situation where this axiom DOESN'T hold. Imagine adding (physically combining) one pile of sand to another pile of sand. You end up with a bigger pile of sand, but it's still ONE pile of sand. Here, 1 + 1 = 1. Does this mean arithmetic's false? Not at all. Instead, it violates the don't-loop axiom, and so it falls outside of the realm of arithmetic. It's akin to wanting to loosen a Phillip's-head screw, and reaching for a ball-peen hammer. You've chosen the wrong tool for the job... and it's your fault, not the tool's.
The first and arguably most-famous axiomatization was Euclid's treatment of geometry from sometime around 300 BCE, which he founded on five postulates. (Meaning axioms. Different name for the same thing.) This was back when an axiom was the same thing in math and philosophy, and Euclid was seriously putting them forward as axioms in the philosophical sense. He formulated all of geometry (as known in his day) as a series of logical deductions derived from these five postulates. In an example of intellectual integrity for the ages, Euclid openly admitted in Elements that he had a lot of doubt about the fifth postulate. He worked very hard to prove everything he could using just the first four postulates, and the stuff that depended on the fifth postulate he flagged as perhaps being unreliable. (The controversy around the fifth postulate is a mathematical epic spanning over two millenia that I won't try to summarize in this post.) Most of the things that Euclid published in Elements were already known, but Euclid's approached the material with a level of mathematical rigor that was nearly unheard-of. Plato's Academy would sport a sign reading "let none ignorant of geometry enter", because geometry after Euclid was seen as the benchmark of logical reasoning, while other areas of math such as arithmetic had not benefited from that rigorous treatment. (In time Greek philosophy would warp itself around geometry. Geometry came to be seen as noble-math and other fields as lesser. Concepts that could not be expressed in Euclidean geometry, such as nullity, negative numbers, the infinite, and the infinitesimal became derided and dismissed as impossibilities. Social philosophies came to be based on geometry, such as nobility being justified by comparison of large similar triangles to smaller ones.)
Let's jump ahead to Renaissance and modern times. Old doctrines were being questioned, including mathematical doctrines. 0 had been imported to Europe with Arabic symbols. Algebra was also imported from the Arab world by way of Spain. Descartes created Analytic Geometry, which essentially took most or all of geometry and boiled it down to algebra -- turning noble math into infidel peasant math. People started playing around with the infinite and the infinitesimal... and Leibniz and/or Newton invented calculus.
Calculus was in particular need of mathematical rigor because no one, not even its inventors, really understood HOW it worked. Newton thought of it in terms of "fluxions" (and the original name was "The Calculus of Fluxions"), where fluxions were variables that would become zero after some mathematical simplification was performed. The problem with this was that it was a cheat. Everyone knew that his technique had no basis in rigorous mathematics. One of his detractors would complain that he was making pacts with the ghosts of numbers departed. Yet at the same time, everyone could tell that his math WORKED.
Around the 19th century, with Calculus becoming increasingly important, other bodies of mathematics emerging, and geometries that called the Euclidean model into serious doubt, a big push was made to reestablish mathematical rigor. Most of our axiomatizations date back to either the 19th or early 20th centuries. The thing is, multiple mathematicians were working on this simultaneously, and they produced multiple different axiomatizations for most of the bodies of mathematics. I've seen about a half-dozen different ways to axiomatize real numbers, to name just one example. Which brings us to the last point I want to make about sets of axioms. In some situations these different axiomiazations lead to subtle differences and we have to be clear which we're working with. (For example, ZF versus ZFC in set theory. The difference between the two, the axiom of choice, is a bright candy shell of "oh obviously that has to be true" covering a sticky caramel center of "WTAF?!?!?!?") In other cases, different axiomatizations produce exactly the same results, even with different axioms. We call these axiomatizations equivalent, and we can prove that Axiomatization A = Axiomatization B by showing that every Axiom of A is a theorem of B and vice-versa. As a result, we can't really say what the axioms of any given body of math are... because most bodies of math have multiple, equivalent formulations of their axioms, and we can't really say that one is any more right than the other because they all amount to the same thing.
BTW, the 19th century's push for rigor ended spectacularly (by mathematician standards of spectacle) with Russel's Paradox. Or as clickbait would describe it, "How Bertrand Russel DESTROYED MATHEMATICS with ONE SIMPLE QUESTION!" But that's a story for another time.
A very, very nice post. I only have a slight quibble about the history.
Plato was a couple of centuries *before* Euclid. So the geometry that Plato wanted for entrance to his Academy was not that of Euclid, but more likely that of Theatetus.
Second, Euclid's Elements *did* include a considerable amount of number theory. For example, his proof that there are infinitely many primes is simple and direct and used today. He also gave a condition when an even number is perfect (later showed to be the only situation where an even number is perfect by Euler).
Given your discussion of the 5th Postulate, I was also surprised that you didn't mention the rise of non-Euclidean geometry and the effect it had on the formalization movement leading up to Russell.
(October 2, 2018 at 7:58 am)polymath257 Wrote: A very, very nice post. I only have a slight quibble about the history.
Plato was a couple of centuries *before* Euclid. So the geometry that Plato wanted for entrance to his Academy was not that of Euclid, but more likely that of Theatetus.
Second, Euclid's Elements *did* include a considerable amount of number theory. For example, his proof that there are infinitely many primes is simple and direct and used today. He also gave a condition when an even number is perfect (later showed to be the only situation where an even number is perfect by Euler).
Given your discussion of the 5th Postulate, I was also surprised that you didn't mention the rise of non-Euclidean geometry and the effect it had on the formalization movement leading up to Russell.
All said, though, well done!
Looks like the forums ate my last attempt at a reply, hopefully this one fairs better.
I didn't mention the number theory in Elements because, while the number theory was also rigorously approached and there was a significant amount of it, it wasn't axiomatized the same way that geometry was. And I didn't mention non-Euclidean geometry because I'd already decided to reserve the 5th Postulate's epic for a potential later post, and also because I didn't feel like spending half the post trying to explain spherical and hyperbolic geometry and how that differs from the Euclidean postulates. And I attributed the Academy's obsession with geometry to Euclid because... .... well, that one was a major screwup on my part. I've believed that one for years but apparently it's false and I never, well, never did the math on that. Good catch.
Being an antipistevist is like being an antipastovist, only with epistemic responsibility instead of bruschetta.
Ooh, I see a competition in math nerdom going on between polymath and Reltzik.
On a different note, good to be back here. I've been thinking of getting a degree in mathematics (with the eventual desire to maybe pursue a higher degree in some advanced math field of interest and become a qualified online tutor or something), but can't seem to find an Australian institution that provides an accredited math program online, and just don't have the time to study on-campus because of my job.
October 2, 2018 at 11:29 am (This post was last modified: October 2, 2018 at 11:42 am by Aliza.)
(October 2, 2018 at 11:05 am)Grandizer Wrote: Ooh, I see a competition in math nerdom going on between polymath and Reltzik.
On a different note, good to be back here. I've been thinking of getting a degree in mathematics (with the eventual desire to maybe pursue a higher degree in some advanced math field of interest and become a qualified online tutor or something), but can't seem to find an Australian institution that provides an accredited math program online, and just don't have the time to study on-campus because of my job.
Yeah, they're talking a bit above my pay grade.
I had a mini-meltdown last night when I couldn't figure out why De Moivre's theorem worked. Like, why can you just take those exponents and drop them all willy nilly into the equation like that? The formula looked familiar enough, but I couldn't for the life of me remember what it was. Surely it's not so simple as Euler's formula. I would have recognized that right away. So I spent all night typing variations of "why the fuck does de moivre's theorem work?" into search engines only to realize that there was a common theme in the answers.
It's Euler's formula.
So yeah, their conversation is a bit over my head. But that's okay! I'm learning.
