The number line freaks me out (2016)
mathwithbaddrawings.com
mathwithbaddrawings.com
The vast majority of people think mathematics is about numbers, when it is actually about relations, and numbers are just some of the entities whose relations mathematics studies.
Nobody is born with this misconception; we teach it, and test it, and thereby ingrain it in the minds of every student, most of whom will never study mathematics at a level that makes them go "wait, what?". The overwhelming majority of people never get to this level.
I suspect this is also why statistics feels so counterintuitive to so many people, including me. The Monty Hall problem is only a problem to those who are naive about probability, which is most people, because most of us don't learn any of this stuff early enough to form long lasting, correct instincts.
It's not fair to students to bake "harmless" lies into their early education, as a way to simplify the topic such that it becomes more easily teachable. We've only done this because teaching is hard, and thus expensive. Education is expensive, at every step. It's not fair or productive to build a gate around proper education that makes it available only to those who can afford it at the level where the early misconceptions get corrected. Even those people end up spending a lot of cognitive capital on all those "wait, what?" moments, when their cognitive capital would be better spent elsewhere.
Well no. Math is studying anything at their most atomic machinations. That mostly involves.
- Making hypothesis, that is assumptions about start conditions and rules of play.
- Evolving the system you just created. Such that conclusions are consistent with the rules of play.
The real deal is good math involves lots of paper work, to an extent you could almost say Math is a writing skill than a thinking skill.
Think of it like generating a lengthy changelog.
Mathematicians have come up with various rules (axioms) that seem to work pretty well. And they spend a great deal of time figuring out their consequences. But it may still happen that the rules have a contradiction and they need to come up with a different set.
Sometimes mathematicians add extra rules when they run into a roadblock. And part of the meta-game is to come up with the minimum set of extra rules they need to keep going. Sometimes they spend time figuring out if the existing rules aren't needed.
Yup, and as you keep going the level too go up!
But the core ideas are simple though-
1. Start some where where you understand things enough to make sense.
2. Make the smallest possible, atomic change to some aspect of thing you know at point 1.
3. Test if the change sticks- If yes, repeat steps 1 - 3
4. If the change doesn't stick- Go back to step 1. Now either make a different change to the same thing or make a new change to a different thing. Repeat steps 1 - 3.
As you can see you write a lot. Like really a lot. Math is just writing skills.
I think it’s more than that… we come with some built-in heuristics for probability, which mostly work pretty well. Until they don’t.
I think of it more as, math is ultimately about symbols. Like, if a mathematician says that "2 = 2" is a true statement, a reasonable onlooker might ask "Does that mean that all twos are interchangeable? Or that there's a unique concept called two and it equals itself?" And the mathematician replies, "Neither! It means that the string of symbols '2 = 2' is reducible to the symbol 'true', given certain axiomatic symbolic transformations. Nothing more, nothing less!".
And obviously we can project concepts onto the symbols, like "integer" and "real number", and talk usefully about them, but those are the map and the symbols are the terrain, as it were. At the edge cases where we're not sure what to think, we have to discard the concepts and consult the symbols.
If that were true, math would be useless, and nothing more than an esoteric artform.
The true power of math comes from the correspondence between those symbolic transformations and observation from the real world. Two objects that look alike can be placed in juxtaposition with any other (different) two objects that look alike, and no matter how much we move them around, as long as we don't add or remove any objects, they can still be placed in the same juxtaposition as before (while this description may seem verbose and clumsy, in the real world it does not need a description - it is a much more primitive sensory perception, learned at an early age).
> obviously we can project concepts onto the symbols, like "integer" and "real number", and talk usefully about them, but those are the map and the symbols are the terrain
It wouldn't be "obvious" that we can project concepts onto symbols, if we didn't discover that symbols correspond to concepts and that symbolic transformations can help us predict the future. Thus I'd say it's the other way around: symbols are the map that we know how to read - of the terrain that we can't traverse easily.
Not at all, think it through further. Obviously it's true that mathematics is more practically useful in cases where its symbolically-proved claims have some kind of relation to real-world observations, but if that relationship were a requirement, math would be useless - you could prove a theory on paper symbolically, but you wouldn't know whether the thing you proved was "really true" until you found a way to check whether the result is also true in the real world. And if you found it was true of apples, it might still not be true for electrons, etc etc.
Rather, math's power stems from the fact that it emphatically does not expect or require the symbols to have any connection to real world observations. If you prove something on paper, it's proved and that's that. If the thing you proved also happens to be useful for describing apples or electrons, that's great - and the fact that this often happens is why the whole "unreasonable effectiveness of mathematics" is a thing. But if there's no relation to the real world, that doesn't in any way affect the truth of the symbolically proved claim, or its usefulness or interest to mathematicians.
What exactly do you mean by "power" here, if not the ability to predict real-world phenomena? In absence of it, what exactly would make it anything more than an exotic artform?
Like, consider: parabolas were pretty fully described by the ancient Greeks, purely as a symbolic abstraction. It was only 1500+ years later that anyone realized that they could also predict the motion of cannonballs and planets. But that discovery was completely orthogonal to the math - e.g. symbolic statements about parabolas didn't get any truer just because they now also described real-world phenomena. (And likewise when we later discovered that planetary motion isn't quite parabolic after all, that didn't affect our understanding of parabolas either.)
That's all I was saying here - that the "esoteric artform" part of math where one abstractly examines symbols is the essence of the thing, and the "predict real-world phenomena" aspect is a side effect that sometimes happens and sometimes doesn't.
You may argue that that, in itself, is powerful, in which case fair enough. But that "power" would be comparable to that of poetry or painting, which, in my opinion, does a disservice to the true power that mathematics holds. Mathematics is much more powerful than poetry and painting, because poetry never helped us build nuclear reactors.
He was absolutely stunned and asked me why mathematics wasn't thought that way all the time. Instead of a bunch of things he had to do, he came to see it as a toolbox with things you can use.
And I myself wonder why the hell my maths teachers failed at making this easier as well. I distinctly remember my math teacher wbo failed to answer me when I asked after months of solving integrals why we need those. I had to figure that out myself, pre-internet.
It is somewhat unfortunate that mathematics is two different things, simultaneously very closely related and very different. One is the abstract study of relationships between axiomatic entities, and the other is arithmetic.
Vast majority of people out there need only arithmetic, and boy they really need it. Calculating tax, taxi fares, shopping bills, splitting bills etc. And to some extent, you need the abstract maths to understand arithmetic.
We have one curriculum for that vast majority of people and for the few who move on to academic maths. Simplifying ideas like integers to number lines doesn't seem like a high price to pay.
Childrens' brains are not fully developed. I see no gain from telling a 6 year old that "most numbers aren't countable". Especially because most numbers are never used or interacted with in any way shape or form. It's not "lying", it's separating concepts and prioritizing.
I mean maybe? Depends on what your definition of being naive about probabilities is. The Monty Hall problem has a sordid history of even very learned mathemathicians specialising in probability getting it very wrong. For example Paul Erdős got it wrong[1] (until someone walked him through it)
Now maybe you count Erdős as someone who is naive about probability. In which case I guess you are right. But that puts the bar very high then.
1: https://sites.oxy.edu/lengyel/M372/Vazsonyi2003/vazs30_1.pdf
That's just silly. We've done that to make the math useful and possible to teach. Unless you're saying you're able to start with sets of numbers and defining a ring for kids, before explaining what 1+1 is.
Thanks but no, we don't need more of this kind of bs naming. "dark matter" already ruined physics because it implies something mysterious and magical is going on whereas it's quite the contrary. I hate it when people dumb down beautiful abstract concepts to the point that it's not only not intuitive, it actually makes the thing less accessible to those who are not in the know.
It's why you can't say e.g. -i < i; the signs on purely imaginary numbers are not an ordering.
Non-symmetric real-valued function on C.
The choice of one as +i and the other as -i is arbitrary, which is not true with 1 and -1.
In any case I’d say this is arbitrary like using + for addition and - for subtraction. It seems like you’re just talking about the symbols themselves. I’m not sure how you get to half plane from there.
(x - x∗) / 2i = 1
where x∗ denotes the complement. If you replace i with -i, the graph will be precisely the complement of the original graph.
For example "i" satisfies the polynomial "x-i=0" and "-i" doesn't. It's just that you can't find any such polynomial with real coefficients that differentiates them.
Of course there are lots of non-algebraic ways to distinguish them too. Or did you mean something stronger?
Complex numbers are generally only in that form when obtained as roots of a polynomial. There are lots of applications where different signs have different interpretations. You can say it's a convention, which is true, but that's not quite the same as saying the two signs are the same thing.
That's always something fun to think about.
Since the set of all English sentences is countable, whereas there are uncountably many real numbers, it follows that there must be numbers that are NOT even definable.
Think about that.
Is it? Where can I read a proof? I have a feeling it’s uncountable set but would be happy to see a proof one way or another.
(The latter statement holds because for any given n, the set X_n of all strings of length n is finite. So you can count the members of X_0, then count the members of X_1, and so on, and by continuing on in that way you'll eventually count out all members of X. You never run out of numbers to assign to the next set because at each point the set of numbers you've already assigned is finite (it's smaller in size than X_0, ..., X_n combined, for some n).
In fact, even if you allow countably infinitely many phonemes to be used, the X_n sets will still be countable, if not finite, and in that case their union is still countable: to see that, you can take enumerations of each set put them together as columns an a matrix. Even though the matrix is infinite in both dimensions, it has finite diagonals, so you can enumerate its cells by going a diagonal at a time, like this (the numbers reflect the cells' order in the numeration):
1 3 6 10 15
2 5 9 14
4 8 13
7 12
11
That just increases the fraction of text files that count as "English". Which doesn't affect the argument.
> the question of whether or not language is finite
does not need to be answered. If English has a thousand words and never gains another one, the list of English sentences is countably infinite. If English gains 10% more words every year forever, the list of English sentences is still countably infinite.
I even provide the definition of countable infinity in my counterargument without realising it, though maybe that too is a misunderstanding.
It feels to me like this is trying to draw an equivalence between language and mathematics yet disallowing the inherent ambiguity of language. At that point, the comparison is just silly.
[1] https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...
This construction by Cauchy sequences looks innocent, but it's the diagonalization argument in disguise. (Start with all the rationals, make sequences out of them, make one that picks one from all, sort them into equivalence classes, and try to map them back to the rationals, notice that you will end up with more equivalence classes.)
The trick is basically that between every rational you can fit an infinite number of irrationals (using the rationals via these sequences). And exactly in this way these are "programs" -- like diagonalization itself. The fact that we can't give programs for most of them is because they are non-computable. (And it's the definition, the indirect proof is above via the cardinalities.)
[but it's dangerously late here, so double check my ramblings ... https://math.stackexchange.com/a/1488502 ]
In a way that makes the real number line continuous. Those numbers have to be there if we want the set to have properties useful for practical applications like algebra.
0.22134967842153005356...
If you get a nondeterministic computer, where every digit it splits into 10 identical computers that each picked one of the 10 options, then when you run that for countably infinite cycles you'll find that you have uncountably infinite computers and you have finally calculated every real number.
The cardinality of the real numbers is 2 to the power of the countable numbers.
Infinity in a box, right in front of your numeric microscope.
Which is why dividing by zero- is exactly the same operation. You take something finite- and you unpack the boxes- in parallel. Every time the operator hits something finite, it unpacks a new set of parallel boxes. The sum of all the boxes, is a infity with a signature.
And those parallel running overlapping infinityssquences, form the irrational numbers
The example the author gives of "fractions" is... rational numbers, and then proceeds to say "what about irrational numbers" - but in mymind (and this is probably where I'm a wrong?) an irrational number is still a fraction of a whole number, just we cannot express it "properly" (yet)
Also, I think I remember that the definition of a rational number implies fractions of integers. Otherwise I could write π as π/1 and give you a rational representation of π.
https://en.wikipedia.org/wiki/Square_root_of_2#Proofs_of_irr...
But the noncomputable numbers make me wonder if our notion of mathematics is too general/powerful.
And when e was defined as a symbol, it was with a computation, (1 + 1/n)^n
If we define "x is computable" as "there exists a Turing machine T(x) which takes n as input and produces n-th digit of x" then there are numbers which are defineable but not computable.
So in a sense math is exploration of the relation between existence and nonexistence.
I've shown you zero fish. The number of fish I've shown you is zero. If I tracked you down, brought a fish with me and showed it to you, I'd have shown you 1 fish, but I haven't.
"The Emperor's New Mind" is a great book on this and related topics.
Right???
In real world, correct, assuming "dot" and "number line" are consist of real world materials.
In math, you need to define "randomly placing a dot" first, because it's proven there isn't a uniform distribution over real numbers ("pick randomly" is usually a colloquial way to say "pick from a uniform distribution.")
It's not impossible, it just has zero probability of occurring.
The video does go further than the article.
And for the line itself, the line is not made up of numbers. Line is made up of continuity, while numbers are cuts in that continuum. Infinite number of cuts do not make up a continuous piece. Mathematical continuity (or extent or measure or span) is the essence of the imaginary spatial existence. It is not composed of cuts. A cut is a non-existence, completely opposite of the existence.
If I pick a number at random using some method for picking that requires me to identify what I picked then 100% of the time I'll get a number I can identify, such as the number that is the solution to x^2=2, or the ratio between a square and a circle, or the quotient of 3 and 7. All those numbers I can't describe will never be picked.
I can do infinitely many coin flips and say the number I picked has the decimals described by that binary sequence. But I'd never be done picking...
Also infinite is not a number. And comparison exists only for numbers.
There are infinitely many integers.
There are infinitely many real numbers between each pair of integers.
Thus there are more real numbers than integers.
Total nitpick, but i think the in in intimidating means "into a state of being timid" and not "in" in the sense of opposite of timidating.