Q: What is it like to understand advanced mathematics? Does it feel analogous to having mastery of another language like in programming or linguistics?
level 0: A state of ignorance. you live in a pre-formal mindset. You don’t know how to formalize things. You don’t even know what it would even mean ‘to prove something mathematically’. This is perhaps the longest. It is the default state of a human. Most anti-theory sentiment comes from this state. Since you’ve neve
You can’t productively read Math books. You often decry that these mathematicians make books way too hard to read. If they only would take the time to explain things simply you would understand.
level 1 : all math is amorphous blob
You know the basic of writing an epsilon-delta proof. Although you don’t know why the rules of maths are this or that way you can at least follow the recipes. You can follow simple short proofs, albeit slowly.
You know there are different areas of mathematics from the unintelligble names in the table of contents of yellow books. They all sound kinda the same to you however.
If you are particularly predisposed to Philistinism you think your current state of knowledge is basically the extent of human knowledge. You will probably end up doing machine learning.
level 2: maths fields diverge
You’ve come so far. You’ve been seriously studying mathematics for several years now. You are proud of yourself and amazed how far you’ve come. You sometimes try to explain math to laymen and are amazed to discover that what you find completely obvious now is complete gibberish to them.
The more you know however, the more you realize what you don’t know. Every time you complete a course you realize it is only scratching the surface of what is out there.
You start to understand that when people talk about concepts in an informal, pre-mathematical way an enormous amount of conceptual issues are swept under the rug. You understand that ‘making things precise’ is actually very difficut.
Different fields of math are now clearly differentiated. The topics and issues that people talk about in algebra, analysis, topology, dynamical systems, probability theory etc wildly differ from each other. Although there are occasional connections and some core conceps that are used all over on the whole specialization is the norm. You realize there is no such thing as a ‘mathematician’: there are logicians, topologists, probability theorist, algebraist.
Actually it is way worse: just in logic there are modal logicians, and set theorist and constructivists and linear logic , and progarmming language people and game semantics.
Often these people will be almost as confused as a layman when they walk into a talk that is supposedly in their field but actually a slightly different subspecialization.
level 3: Galactic Brain of Percolative Convergence
As your knowledge of mathematics you achieve the Galactic Brain take level of percolative convergence: the different fields of mathematics are actually highly interrelated—the connections percolate to make mathematics one highly connected component of knowledge.
You are no longer suprised on a meta level to see disparate fields of mathematics having unforeseen & hidden connections—but you still appreciate them.
You resist the reflexive impulse to divide mathematics into useful & not useful—you understand that mathematics is in the fullness of Platonic comprehension one unified discipline. You’ve taken a holistic view on mathematics—you understand that solving the biggest problems requires tools from many different toolboxes.
I say that knowing particular kinds of math, the kind that let you model the world more-precisely, and that give you a theory of error, isn’t like knowing another language. It’s like knowing language at all. Learning these types of math gives you as much of an effective intelligence boost over people who don’t, as learning a spoken language gives you above people who don’t know any language (e.g., many deaf-mutes in earlier times).
The kinds of math I mean include:
how to count things in an unbiased manner; the methodology of polls and other data-gathering
how to actually make a claim, as opposed to what most people do, which is to make a claim that’s useless because it lacks quantification or quantifiers
A good example of this is the claims in the IPCC 2015 report that I wrote some comments on recently. Most of them say things like, “Global warming will make X worse”, where you already know that OF COURSE global warming will make X worse, but you only care how much worse.
More generally, any claim of the type “All X are Y” or “No X are Y”, e.g., “Capitalists exploit the working class”, shouldn’t be considered claims at all, and can accomplish nothing except foment arguments.
the use of probabilities and error measures
probability distributions: flat, normal, binomial, poisson, and power-law
entropy measures and other information theory
predictive error-minimization models like regression
statistical tests and how to interpret them
These things are what I call the correct Platonic forms. The Platonic forms were meant to be perfect models for things found on earth. These kinds of math actually are. The concept of “perfect” actually makes sense for them, as opposed to for Earthly categories like “human”, “justice”, etc., for which believing that the concept of “perfect” is coherent demonstrably drives people insane and causes them to come up with things like Christianity.
They are, however, like Aristotle’s Forms, in that the universals have no existence on their own, but are (like the circle , but even more like the normal distribution ) perfect models which arise from the accumulation of endless imperfect instantiations of them.
There are plenty of important questions that are beyond the capability of the unaided human mind to ever answer, yet which are simple to give correct statistical answers to once you know how to gather data and do a multiple regression. Also, the use of these mathematical techniques will force you to phrase the answer sensibly, e.g., “We cannot reject the hypothesis that the average homicide rate under strict gun control and liberal gun control are the same with more than 60% confidence” rather than “Gun control is good.”
If they only would take the time to explain things simply you would understand.
This is an interesting one. I field this comment quite often from undergraduates, and it’s hard to carve out enough quiet space in a conversation to explain what they’re doing wrong. In a way the proliferation of math on YouTube might be exacerbating this hard step from tourist to troubadour.
The Vibes of Mathematics:
Q: What is it like to understand advanced mathematics? Does it feel analogous to having mastery of another language like in programming or linguistics?
A: It’s like being stranded on a tropical island where all your needs are met, the weather is always perfect, and life is wonderful.
Except nobody wants to hear about it at parties.
Vibes of Maths: Convergence and Divergence
level 0: A state of ignorance. you live in a pre-formal mindset. You don’t know how to formalize things. You don’t even know what it would even mean ‘to prove something mathematically’. This is perhaps the longest. It is the default state of a human. Most anti-theory sentiment comes from this state. Since you’ve neve
You can’t productively read Math books. You often decry that these mathematicians make books way too hard to read. If they only would take the time to explain things simply you would understand.
level 1 : all math is amorphous blob
You know the basic of writing an epsilon-delta proof. Although you don’t know why the rules of maths are this or that way you can at least follow the recipes. You can follow simple short proofs, albeit slowly.
You know there are different areas of mathematics from the unintelligble names in the table of contents of yellow books. They all sound kinda the same to you however.
If you are particularly predisposed to Philistinism you think your current state of knowledge is basically the extent of human knowledge. You will probably end up doing machine learning.
level 2: maths fields diverge
You’ve come so far. You’ve been seriously studying mathematics for several years now. You are proud of yourself and amazed how far you’ve come. You sometimes try to explain math to laymen and are amazed to discover that what you find completely obvious now is complete gibberish to them.
The more you know however, the more you realize what you don’t know. Every time you complete a course you realize it is only scratching the surface of what is out there.
You start to understand that when people talk about concepts in an informal, pre-mathematical way an enormous amount of conceptual issues are swept under the rug. You understand that ‘making things precise’ is actually very difficut.
Different fields of math are now clearly differentiated. The topics and issues that people talk about in algebra, analysis, topology, dynamical systems, probability theory etc wildly differ from each other. Although there are occasional connections and some core conceps that are used all over on the whole specialization is the norm. You realize there is no such thing as a ‘mathematician’: there are logicians, topologists, probability theorist, algebraist.
Actually it is way worse: just in logic there are modal logicians, and set theorist and constructivists and linear logic , and progarmming language people and game semantics.
Often these people will be almost as confused as a layman when they walk into a talk that is supposedly in their field but actually a slightly different subspecialization.
level 3: Galactic Brain of Percolative Convergence
As your knowledge of mathematics you achieve the Galactic Brain take level of percolative convergence: the different fields of mathematics are actually highly interrelated—the connections percolate to make mathematics one highly connected component of knowledge.
You are no longer suprised on a meta level to see disparate fields of mathematics having unforeseen & hidden connections—but you still appreciate them.
You resist the reflexive impulse to divide mathematics into useful & not useful—you understand that mathematics is in the fullness of Platonic comprehension one unified discipline. You’ve taken a holistic view on mathematics—you understand that solving the biggest problems requires tools from many different toolboxes.
I say that knowing particular kinds of math, the kind that let you model the world more-precisely, and that give you a theory of error, isn’t like knowing another language. It’s like knowing language at all. Learning these types of math gives you as much of an effective intelligence boost over people who don’t, as learning a spoken language gives you above people who don’t know any language (e.g., many deaf-mutes in earlier times).
The kinds of math I mean include:
how to count things in an unbiased manner; the methodology of polls and other data-gathering
how to actually make a claim, as opposed to what most people do, which is to make a claim that’s useless because it lacks quantification or quantifiers
A good example of this is the claims in the IPCC 2015 report that I wrote some comments on recently. Most of them say things like, “Global warming will make X worse”, where you already know that OF COURSE global warming will make X worse, but you only care how much worse.
More generally, any claim of the type “All X are Y” or “No X are Y”, e.g., “Capitalists exploit the working class”, shouldn’t be considered claims at all, and can accomplish nothing except foment arguments.
the use of probabilities and error measures
probability distributions: flat, normal, binomial, poisson, and power-law
entropy measures and other information theory
predictive error-minimization models like regression
statistical tests and how to interpret them
These things are what I call the correct Platonic forms. The Platonic forms were meant to be perfect models for things found on earth. These kinds of math actually are. The concept of “perfect” actually makes sense for them, as opposed to for Earthly categories like “human”, “justice”, etc., for which believing that the concept of “perfect” is coherent demonstrably drives people insane and causes them to come up with things like Christianity.
They are, however, like Aristotle’s Forms, in that the universals have no existence on their own, but are (like the circle , but even more like the normal distribution ) perfect models which arise from the accumulation of endless imperfect instantiations of them.
There are plenty of important questions that are beyond the capability of the unaided human mind to ever answer, yet which are simple to give correct statistical answers to once you know how to gather data and do a multiple regression. Also, the use of these mathematical techniques will force you to phrase the answer sensibly, e.g., “We cannot reject the hypothesis that the average homicide rate under strict gun control and liberal gun control are the same with more than 60% confidence” rather than “Gun control is good.”
Thanks for writing this. I only wish it was longer.
You seem to do OK…
This is an interesting one. I field this comment quite often from undergraduates, and it’s hard to carve out enough quiet space in a conversation to explain what they’re doing wrong. In a way the proliferation of math on YouTube might be exacerbating this hard step from tourist to troubadour.