Why was the axiomatic method never taught to me?
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Why was the axiomatic method never taught to me?
It is best if I start by describing myself.
Ever since I was in 1st grade, maybe kindergarten, I know that I wanted to be a scientist. The exact kind and field changed over the years, but that love of science was always constant throughout. I easily passed every science and math class my school gave me. On my senior year, I took English (4 years was required), Law & Society (the closest thing my school had to philosophy or logic), 2 studies (the max. allowed by administration) and 5 sciences (Robotics, Destination Imagination, Physics, Disease Diagnostic a.k.a. House, and Programming I & II).
The point I want to make is that I did not miss the lessons on the axiomatic system because I did not take the advanced classes or because I was not smart enough to realize when a teacher was talking about it subtly.
__________________________________________________________________________________________________________
During freshman geometry, I learned about the axiomatic system that math is based on and the 5 postulated Euclid used. After the week, I never heard about the system again. A teacher never went of on a tangent about it or made a passing, offhand comment. It was like how no teacher dared to talk about religion (remember, it was a public school). The lack of discussing about something so important was to major to be accidental. Even the smart people on YouTube do not mention it. Go look at the uploads by MinutePhysics/MinuteEarth, Veritasium, VSauce, Sixty Symbols etc. I have watched the vast majority of those videos and they never mention it.
The axiomatic system is what makes math math. The only two systems of reasoning science uses is the axiomatic system and the empirical system. Logic, by its definition, is an axiomatic system. There is no way around it, this is something too important to not be taught. Yet its absence is everywhere.
I can only think of two reasons why the axiomatic theory was left out. The first is that the teachers choose to hide it from their students. The only reason I can think of for not teaching the axiomatic system is that it tells the student that science and math are fragile, that they are not infallible and that ultimate truth, truth that is, was and will be, cannot be reached by humans. The second is that they did not know about it. Honestly, I cannot decide which is worse.
I am sorry if this is just me rambbleing and ranting. I started this post at 6 AM or so and... well... you know how it goes (http://xkcd.com/68/)
Ever since I was in 1st grade, maybe kindergarten, I know that I wanted to be a scientist. The exact kind and field changed over the years, but that love of science was always constant throughout. I easily passed every science and math class my school gave me. On my senior year, I took English (4 years was required), Law & Society (the closest thing my school had to philosophy or logic), 2 studies (the max. allowed by administration) and 5 sciences (Robotics, Destination Imagination, Physics, Disease Diagnostic a.k.a. House, and Programming I & II).
The point I want to make is that I did not miss the lessons on the axiomatic system because I did not take the advanced classes or because I was not smart enough to realize when a teacher was talking about it subtly.
__________________________________________________________________________________________________________
During freshman geometry, I learned about the axiomatic system that math is based on and the 5 postulated Euclid used. After the week, I never heard about the system again. A teacher never went of on a tangent about it or made a passing, offhand comment. It was like how no teacher dared to talk about religion (remember, it was a public school). The lack of discussing about something so important was to major to be accidental. Even the smart people on YouTube do not mention it. Go look at the uploads by MinutePhysics/MinuteEarth, Veritasium, VSauce, Sixty Symbols etc. I have watched the vast majority of those videos and they never mention it.
The axiomatic system is what makes math math. The only two systems of reasoning science uses is the axiomatic system and the empirical system. Logic, by its definition, is an axiomatic system. There is no way around it, this is something too important to not be taught. Yet its absence is everywhere.
I can only think of two reasons why the axiomatic theory was left out. The first is that the teachers choose to hide it from their students. The only reason I can think of for not teaching the axiomatic system is that it tells the student that science and math are fragile, that they are not infallible and that ultimate truth, truth that is, was and will be, cannot be reached by humans. The second is that they did not know about it. Honestly, I cannot decide which is worse.
I am sorry if this is just me rambbleing and ranting. I started this post at 6 AM or so and... well... you know how it goes (http://xkcd.com/68/)
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Re: Why was the axiomatic method never taught to me?
The five Euclidean axioms are what geometry is based on, not what all of math is based on. It may be possible to base other fields of mathematics on Euclidean geometry (and it wouldn't surprise me if someone has done so), but, at least in modern times, from my understanding, it's more typical to base all of math on set theory. Other fields have their own axioms (or definitions, which are also unprovable starting points); for instance, arithmetic has the Peano axioms; and group theory, at least in the class I took, had the definition of a group fairly early on, which is sort of like the axioms for that particular field.
As to why you don't hear about axioms that much: because it's not always that interesting or useful. Deriving everything from basic axioms takes a lot of time and effort (analogous to how people don't program in assembly language all the time), and there's a lot of stuff that you can do without using the axioms directly (but rather, using results that other people have already proven). And in many cases, the axioms aren't really how people think about the topic. If the class is about applied math, rather than pure math, then talking about the axioms explicitly isn't that useful; any science will be applied math, and probably most high school class will be, too (as well as at least some college classes, particularly those aimed at non–math majors).
As to why you don't hear about axioms that much: because it's not always that interesting or useful. Deriving everything from basic axioms takes a lot of time and effort (analogous to how people don't program in assembly language all the time), and there's a lot of stuff that you can do without using the axioms directly (but rather, using results that other people have already proven). And in many cases, the axioms aren't really how people think about the topic. If the class is about applied math, rather than pure math, then talking about the axioms explicitly isn't that useful; any science will be applied math, and probably most high school class will be, too (as well as at least some college classes, particularly those aimed at non–math majors).
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Re: Why was the axiomatic method never taught to me?
Seconded. It's kind of neat, but it's not particularly useful.chridd wrote:As to why you don't hear about axioms that much: because it's not always that interesting or useful. Deriving everything from basic axioms takes a lot of time and effort (analogous to how people don't program in assembly language all the time), and there's a lot of stuff that you can do without using the axioms directly (but rather, using results that other people have already proven). And in many cases, the axioms aren't really how people think about the topic.
I only learned about Euclid's axioms in passing when my linear algebra teacher (technically firstyear undergrad) mentioned it in passing; he was the sort to digress onto side topics like that. I probably never would have heard of them otherwise.
We've had long rambling threads here in the past in which people contemplate the ways they think math ought to be taught – see for instance viewtopic.php?t=56231&p=2002531 , or viewtopic.php?f=44&t=35795 , or viewtopic.php?f=44&t=35311 . My opinion remains that it might sound good and pure and wholesome to try to instill in students some inspiring view of the beauty of mathematics, but such an approach will not readily work for every professor and every classroom and it's best to try to stick to the essentials (whatever they are).
These don't sound particularly sciencey.5 sciences (Robotics, Destination Imagination, Physics, Disease Diagnostic a.k.a. House, and Programming I & II).
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 doogly
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Re: Why was the axiomatic method never taught to me?
Science is never axiomatic, that is anathema to us. You can find something like "axioms of quantum mechanics" in textbooks, but that is because we want our models, although developed empirically, to be stated parsimoniously.
Much of math is also the same way. A weird historical quirk is that spherical trigonometry was worked out for centuries, due to its use in navigation and astronomy, prior to the 19th century work on axiomatic spherical geometry through alternatives to Euclid's fifth postulate. In some fields the axiomatic system motivates progress, but it's often just "tidying." Or, a revisionist history, a la Bourbaki, where the presentation proceeds along completely separate logical lines than the path of discovery. Much of the status of "foundational mathematics" is actually quite independent. Let's say some flaw or surprise were discovered with set theory  what would happen to our understanding of complex analysis? In most cases, the answer is very little. In some cases, like in algebraic geometry, we do have some interesting hinges, like with the AOC. But if you're living in a house and the foundation goes, you need to move out of this death trap and hand your savings over to a contractor, because you are in severe danger. In math, you will need to maybe twerk a little language, or just leave it to the colleagues who care. In science, it would have been deeply wrong to care in the first place.
Much of math is also the same way. A weird historical quirk is that spherical trigonometry was worked out for centuries, due to its use in navigation and astronomy, prior to the 19th century work on axiomatic spherical geometry through alternatives to Euclid's fifth postulate. In some fields the axiomatic system motivates progress, but it's often just "tidying." Or, a revisionist history, a la Bourbaki, where the presentation proceeds along completely separate logical lines than the path of discovery. Much of the status of "foundational mathematics" is actually quite independent. Let's say some flaw or surprise were discovered with set theory  what would happen to our understanding of complex analysis? In most cases, the answer is very little. In some cases, like in algebraic geometry, we do have some interesting hinges, like with the AOC. But if you're living in a house and the foundation goes, you need to move out of this death trap and hand your savings over to a contractor, because you are in severe danger. In math, you will need to maybe twerk a little language, or just leave it to the colleagues who care. In science, it would have been deeply wrong to care in the first place.
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Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Re: Why was the axiomatic method never taught to me?
I can think of two reasons: One, it's so far in the realm of pure mathematical theory, separated from practical application, that very few people other than professional mathematicians have any reason to care about it. Two, it's such a basic fundamental foundation that people generally get an instinctive feel for the parts of it that matter to them without ever needing to be taught it formally.
I'm not sure exactly what you're talking about, but there are a lot of things in math that if you taught them, most people would respond with either "duh, why are you even bothering to mention this, it's so obvious?" or "how could that possibly be relevant to me?" This is true even if the people in question are studying fairly advanced mathematical topics such as calculus and differential equations. What you describe sounds very much like it is such a thing. It may be incredibly important in the widespread implications it has for many branches of mathematics, but it is those implications that are important and the relationship between the axioms and the implications may be far from obvious or short. It is sufficient, and much easier, to teach most people just the implications, which usually stand on their own pretty well.
I'm not sure exactly what you're talking about, but there are a lot of things in math that if you taught them, most people would respond with either "duh, why are you even bothering to mention this, it's so obvious?" or "how could that possibly be relevant to me?" This is true even if the people in question are studying fairly advanced mathematical topics such as calculus and differential equations. What you describe sounds very much like it is such a thing. It may be incredibly important in the widespread implications it has for many branches of mathematics, but it is those implications that are important and the relationship between the axioms and the implications may be far from obvious or short. It is sufficient, and much easier, to teach most people just the implications, which usually stand on their own pretty well.
Re: Why was the axiomatic method never taught to me?
Rhombic wrote:Gödel.
From the sound of it, you [the OP] probably haven't run into Gödel yet (or at least just recently googled to find out what that means). Basically, axiomatic methods have been important in math roughly twice: First with nonEuclidian math proving that Euclid's axioms weren't the only possible axioms (and also that none of them can apparently be proved by the rest). Second when Kurt Gödel proved that any formal system (one of the requirements was to be rigidly dependent on said axioms) would not be able to prove all true conjectures.
An even bigger reason that axiomatic methods aren't taught outside of geometry is that the system Gödel was working with, Principia Mathematica, is a bit of a nightmare and certainly not expected for undergraduates (it doesn't derive multiplication until somewhere after page 100). Working out the axioms needed for Algebra (and anything built on Algebra) is nontrivial. Learning to use such a system isn't much easier.
Finally, note that which axioms should be used, and which might generate contradictions isn't quite clear. Consider the Axiom of Choice and Banach–Tarski paradox.
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Re: Why was the axiomatic method never taught to me?
Axioms are not Foundational.
Formal Mathematics is a game.
You have symbols, and rules for how these symbols are allowed to be written out next to each other.
In one sense, Axioms are just legal starting spots, and Derivation rules are the rules about how you can chain together stuff.
As it happens, it is convincing that if we start with Axioms whose interpretation is true, and we pick Derivation rules that preserve truth, we can play this game and generate other formal statements, interpret them, and get true interpretations.
The fun thing is you can play this game with really formal mathematics, or less formal mathematics. So long as you are reasonable careful with interpretation, the "seed" statements (axioms), and the derivation rules as being propertypreserving, you can play in this abstract game space and generate useful information outside of it.
A really basic example of this game is counting. An allowed interpretation of the number 7 is that it talks about 7 of something, addition as bringing together, and subtraction as taking away.
So if we have two groups of sheep, and one of them aligns with the interpretation of 7, and the other with the interpretation of 3, we can combine them (add) to get a group of 10 sheep. Really basic grade school stuff, but it is the game of mathematics.
Instead of having to think about the sheep individually (the one with the floppy ear, lame leg, grey coat, etc), we can map to numbers, play with numbers, then map back to sheep. I know I can break those 10 sheep up into 5 groups of 2.
Now, once my interpretation breaks down  maybe when you combine the two groups, the sheep don't like each other, so one of the sheep is killed  then the game is less useful, or I need to play a more complex game.
Calculus is just this game writ larger. If we have things that behave reasonably like smooth functions, we can do calculus on the smooth functions and get out results that can be aligned with the corresponding combination of the things. You can even describe how far from said smooth functions the things can be such that the result is still useful (so calculus when reality is "actually" granular on the Planck scale remains useful).
As it happens, this playing of games is fun in and of itself. We take systems, combine simplify or otherwise modify them, and see what results. We find some thing we wouldn't mind knowing, and try to figure out what game system makes it knowable. We find properties (like primes) of useful systems (like counting numbers) and play with them to see if anything falls out (like cryptography).
These games that interest us tend to be about things that are easy to reason about in a very vague way. When we play games on raw chaos, we reason about what patterns fall out. Even most extremely abstract mathematics is about the spots where there is structure (patterns we can understand) to sink our teeth into.
Such abstractions end up being useful more often than one might expect, even if we started without any practical application. Because once we find some neat properties, we can search for things for which this game corresponds to (or force it), and get an explosion of information from the interpretation of relatively basic things into our mathematical game.
But the axioms are not the foundation.
If we have things we correspond to a game (with axioms and derivation rules), and we find some neat things that are true in the game (and hence true about the things), if it turns out the game we are playing isn't a good game (it is inconsistent, for example), that doesn't cost us the truths we have discovered about the things that we verified independently of the game.
We even build games on top of other games (calculus on top of set theory, for example). The calculus game does not depend on the set theory game, despite being able to derive the calculus game from set theory.
Our knowledge of physics doesn't depend on the calculus game being a good game. The calculus game gave us a way to spread truth. If it is a bad game, some of the ways it spreads truth are not valid. But we check the game against physics: all it might do is provide us with clues where to look, and it is useful. If our particular calculus game turns out to be junk, we can invent a new game and play physics with it.
The axioms are not foundational, the game and correspondence between games is.
As you go through a mathematical and scientific education, you are taught various games. How to take equations and balance them, how to solve for x, etc. There are low level axiom systems with derivation rules that can be used to derive those games and their rules using formal mathematics, but that low level game isn't the reason why the higher level game works.
The game of solving for x has independent usefulness without the axioms of set theory. And you where taught that game and how to play it. You where not taught axioms and the like, because those really don't help you play the game of solving for x.
Now, formal axiom type math games have uses. They reduce what things you have to believe to be true in order for the interpretation game and derivation game to produce convincing results. So you can reduce more complex games to such "simpler" games and convince yourself those more complex games are sound, then continue playing the more complex games without having to go back to the low level game.
At one point, there was a hope to find the game to end all games. The one universal game from which you can derive the play all other good games. Kurt Gödel proved, by playing a game, that no game is universal: formal logic systems are either weak, incomplete or inconsistent.
Which means we have to invent useful games when we find a use for them, or just because it is fun.
Formal Mathematics is a game.
You have symbols, and rules for how these symbols are allowed to be written out next to each other.
In one sense, Axioms are just legal starting spots, and Derivation rules are the rules about how you can chain together stuff.
As it happens, it is convincing that if we start with Axioms whose interpretation is true, and we pick Derivation rules that preserve truth, we can play this game and generate other formal statements, interpret them, and get true interpretations.
The fun thing is you can play this game with really formal mathematics, or less formal mathematics. So long as you are reasonable careful with interpretation, the "seed" statements (axioms), and the derivation rules as being propertypreserving, you can play in this abstract game space and generate useful information outside of it.
A really basic example of this game is counting. An allowed interpretation of the number 7 is that it talks about 7 of something, addition as bringing together, and subtraction as taking away.
So if we have two groups of sheep, and one of them aligns with the interpretation of 7, and the other with the interpretation of 3, we can combine them (add) to get a group of 10 sheep. Really basic grade school stuff, but it is the game of mathematics.
Instead of having to think about the sheep individually (the one with the floppy ear, lame leg, grey coat, etc), we can map to numbers, play with numbers, then map back to sheep. I know I can break those 10 sheep up into 5 groups of 2.
Now, once my interpretation breaks down  maybe when you combine the two groups, the sheep don't like each other, so one of the sheep is killed  then the game is less useful, or I need to play a more complex game.
Calculus is just this game writ larger. If we have things that behave reasonably like smooth functions, we can do calculus on the smooth functions and get out results that can be aligned with the corresponding combination of the things. You can even describe how far from said smooth functions the things can be such that the result is still useful (so calculus when reality is "actually" granular on the Planck scale remains useful).
As it happens, this playing of games is fun in and of itself. We take systems, combine simplify or otherwise modify them, and see what results. We find some thing we wouldn't mind knowing, and try to figure out what game system makes it knowable. We find properties (like primes) of useful systems (like counting numbers) and play with them to see if anything falls out (like cryptography).
These games that interest us tend to be about things that are easy to reason about in a very vague way. When we play games on raw chaos, we reason about what patterns fall out. Even most extremely abstract mathematics is about the spots where there is structure (patterns we can understand) to sink our teeth into.
Such abstractions end up being useful more often than one might expect, even if we started without any practical application. Because once we find some neat properties, we can search for things for which this game corresponds to (or force it), and get an explosion of information from the interpretation of relatively basic things into our mathematical game.
But the axioms are not the foundation.
If we have things we correspond to a game (with axioms and derivation rules), and we find some neat things that are true in the game (and hence true about the things), if it turns out the game we are playing isn't a good game (it is inconsistent, for example), that doesn't cost us the truths we have discovered about the things that we verified independently of the game.
We even build games on top of other games (calculus on top of set theory, for example). The calculus game does not depend on the set theory game, despite being able to derive the calculus game from set theory.
Our knowledge of physics doesn't depend on the calculus game being a good game. The calculus game gave us a way to spread truth. If it is a bad game, some of the ways it spreads truth are not valid. But we check the game against physics: all it might do is provide us with clues where to look, and it is useful. If our particular calculus game turns out to be junk, we can invent a new game and play physics with it.
The axioms are not foundational, the game and correspondence between games is.
As you go through a mathematical and scientific education, you are taught various games. How to take equations and balance them, how to solve for x, etc. There are low level axiom systems with derivation rules that can be used to derive those games and their rules using formal mathematics, but that low level game isn't the reason why the higher level game works.
The game of solving for x has independent usefulness without the axioms of set theory. And you where taught that game and how to play it. You where not taught axioms and the like, because those really don't help you play the game of solving for x.
Now, formal axiom type math games have uses. They reduce what things you have to believe to be true in order for the interpretation game and derivation game to produce convincing results. So you can reduce more complex games to such "simpler" games and convince yourself those more complex games are sound, then continue playing the more complex games without having to go back to the low level game.
At one point, there was a hope to find the game to end all games. The one universal game from which you can derive the play all other good games. Kurt Gödel proved, by playing a game, that no game is universal: formal logic systems are either weak, incomplete or inconsistent.
Which means we have to invent useful games when we find a use for them, or just because it is fun.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
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Re: Why was the axiomatic method never taught to me?
Damn, son. You got game.
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 doogly
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Re: Why was the axiomatic method never taught to me?
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Re: Why was the axiomatic method never taught to me?
... I'm gonna go repost that in another thread that could certainly use it.
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Re: Why was the axiomatic method never taught to me?
All of the above posts are excellent and plausible explanations as to why the OP was never taught the axiomatic method but the real reason is conspiratorial. We here at the Ordo Trismegistus determined long ago that jewish_scientist must never learn that the Method exists. You might say that that is an axiom with our Society and, now that the cat is out of the bag, we shall have to rectify the situation.
Estragon: I can't go on like this.
Vladimir: That's what you think.
Vladimir: That's what you think.
Re: Why was the axiomatic method never taught to me?
An excellent analogy some mathematician (I want to say Ian Stewart, but I could be wrong) once put forward (note, I'm paraphrasing from memory, so the details will probably be wrong, though the general gist and possibly also the style should be right):
If maths is like a building, then axioms are the foundations, way down below the subbasement. Or maybe they're the bedrock the foundations rest on. Either way, most mathematicians would agree that, like foundations, it's very important to have solid axioms, otherwise your walls are liable to collapse, and your roof fall in. However, unlike buildings, where it's generally agreed that you should start with the foundations and build up from there, real mathematicians build their structures starting with the top  if maths is like a building, then mathematicians start by tiling the roof, then put some roofbeams in place, and only once they're sure they've got a roof they like the look of do they start putting in some scaffolding in place of walls, that might even reach the ground in places  more likely it skews sideways and lights on someone else's roof. You can have vast, interconnected structures built on each other like some weird Escher construction, that noone notices haven't yet reached the ground, let alone having support from the deep foundations. Eventually, someone comes along who's interested in the walls and puts in some solid support columns, and there are miners deep underground, digging down to establish ever deeper foundations, but most of the maths  the daytoday work of most mathematicians  is on the tenuously supported rooftops.
Real mathematicians know you build from the top down, and only put in the walls once you've got a sturdy roof.
If maths is like a building, then axioms are the foundations, way down below the subbasement. Or maybe they're the bedrock the foundations rest on. Either way, most mathematicians would agree that, like foundations, it's very important to have solid axioms, otherwise your walls are liable to collapse, and your roof fall in. However, unlike buildings, where it's generally agreed that you should start with the foundations and build up from there, real mathematicians build their structures starting with the top  if maths is like a building, then mathematicians start by tiling the roof, then put some roofbeams in place, and only once they're sure they've got a roof they like the look of do they start putting in some scaffolding in place of walls, that might even reach the ground in places  more likely it skews sideways and lights on someone else's roof. You can have vast, interconnected structures built on each other like some weird Escher construction, that noone notices haven't yet reached the ground, let alone having support from the deep foundations. Eventually, someone comes along who's interested in the walls and puts in some solid support columns, and there are miners deep underground, digging down to establish ever deeper foundations, but most of the maths  the daytoday work of most mathematicians  is on the tenuously supported rooftops.
Real mathematicians know you build from the top down, and only put in the walls once you've got a sturdy roof.
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