What Even Is a Number?
notebook.drmaciver.com
notebook.drmaciver.com
OK, I said, show me a plain old "one". Nope, that's one fork; nope that's one ball; nope, you get the idea.
You cannot show just "one" unattached to anything else.
The concept of "one" is an idea. It only exists as a "real" concept in your mind because it exists as the same concept in the minds of others, and it's a very useful concept in describing the world around us.
This odd fact becomes ever more apparent when you find out about cultures that have very different approaches to quantitative reasoning, like the Pirahã in Brazil [1]. They don't have number words beyond "one", words roughly meaning "some" and "many" are used for anything more than one.
[1] https://www.sciencedaily.com/releases/2008/07/080714111940.h...
Ie., you're asking someone to find you an object with spatio-temporal properties which, as an object, does not have spatio-temporal properties.
Likewise you could ask, "find me a smile?!!" and then deny every example was a smile because "it was only a face smiling!".
"Existence" is defined relative to a context in which it can be evaluated, and it -- in general -- does not imply any particular properties.
"To Exit" in general is "to be found in some domain"
2 can be found in the domain of numbers
I probably should, it'd be fun and they're a bit older than when I pulled the "one" game.
When we reason about numbers, we're embedding a representation of a universe of numbers into our universe.
What we call "physical existence" could is also probably just be "math all the way down". Just not in such a way that the number two per se can be an entity for us to behold.
How is it any different than believing that unicorns exist as physical objects, just in their own universe?
What we call "physical existence" could...just be "math all the way down".
Or, math is also a number of simplified alternate views of reality of varying degrees of "applicability." (Where "applicability" has to do with predictive power over reality.)
We have very well-developed representations of the number universe that we can embed into our own universe. These things are governed by precise axioms. By contrast, we don't have such well-developed "unicorn universe".
(For starters, what is the definition of "unicorn"? If it's just a horse with a horn, then such a thing is plausible with genetic engineering in our universe.) As we endow the unicorn with additional properties, then it becomes less and less clear that there exists a universe where such a thing can exists other than as an imaginary being. If the concept of a unicorn could be axiomatized, then it could constitute a universe.
> Or, math is also a number of simplified alternate views of reality.
Well, that's the math that we do; not the math that we (possibly) are.
Indeed. The current better known attempts at a "unicorn universe" have problems with power scaling and the square-cube law. https://www.reddit.com/r/mylittlepony/comments/mrs81/accordi...
Do you have a good definition of what constitutes a "well-developed" universe?
An easy definition would be one where everyone agrees on the exact mechanics and sequence of states. The fibonacci sequence is well-developed: it has an exact definition that every competent simulator can simulate the same way. If I say my favorite number is the 1000th item in the sequence (starting with 0, 1, 1 as items #1, #2, #3), then everyone can calculate the 1000th item the same way.
If we all try to talk about what happens in the "unicorn universe", everyone has their own different ideas about that. There's no unambiguous rules for how the unicorn universe is calculated. Everyone has a different "unicorn universe" in their heads, and each of those universes operates on our own mind's whims rather than any rules we can describe to each other.
Homo sapiens has this issue with this universe. Even if you subset that group to just scientists, there are still differences of opinion.
Good question, but I think there might be a difference. Suppose, in a galaxy far, far away, there are aliens. They probably don't have the concept of unicorns. However, they probably have discovered numbers, pi, Pythagoras theorem, e^(i pi) + 1 = 0, etc.
So, does the fact that something is discovered (rather than invented), and can be discovered many times independently in identical fashion, confer it some sort of elevated ontological status?
Rather: if we have a notion of something and statements about it in every universe, then it must exist. However, come to think of it, that's bonkers.
Still, it's remarkable that mathematics allows us to make reproducible true statements about fictional objects.
We can only be confident of that if they're in this universe. Also, this is only a conjecture. It's just something some of us imagine would be true. You can't elevate such an idea to the level of empirical data.
So, does the fact that something is discovered (rather than invented), and can be discovered many times independently in identical fashion, confer it some sort of elevated ontological status?
Unicorns. Kirin. Narwhals. Rhinos. So you have two independent origins in culture for such a creature, plus two appearances of something similar in nature. Does that confer it some sort of elevated ontological status?
It's not at all clear that the cultural constructs are independent of each other or real world creatures. Particularly in the Western case, some of the earliest Greek accounts of the unicorn as well as Marco Polo’s account confirming its existence and difference from the contemporary evolution of the image stem from the distant East (from the European perspective) and clearly are describing a rhinoceros. They are also very similar to accounts of the Kirin.
What would happen to your epistemology if every alien civilization has unicorns but some lack π?
I knew it, Unicorns are real!!11!
I am not sure I can conceive of aliens that don't have pi (how can they have cylindrical rockets?).
Anyway, maybe we should let the professionals deal with this.
https://plato.stanford.edu/entries/platonism-mathematics/
https://plato.stanford.edu/entries/intuitionism/
https://plato.stanford.edu/entries/mathphil-indis/
You could likewise argue that sounds exist in a different universe, but are only "embedded" into the air using pressure waves, but why would you ever think you needed a different universe of sounds when you've already got pressure waves in air? I mean... what's "not enough" about that for you?
>What we call "physical existence" could is also probably just be "math all the way down".
This is a needless diversion. Mathematics is the study of structure. Physical reality has structure, and very well could "be structure" in some equivalent sense, but to say it is math is just abject poetic nonsense. This change of language would literally make no significant difference to anybody about anything. You may as well be declaring the world to be flat and asking everybody else to go and change all our language to make it true.
(personally I think mathematics exists in our brains, but also and crucially in the interactions between our brains, since a proof is never accepted until it is reviewed and so on. A proof that's just in my brain is no proof at all, even if it happens to be correct)
Besides, there are plenty of mathematical concepts which are hard to understand. I don't see any uniqueness to math here.
No it isn't! Either it is math all the way down, or else it isn't.
If it isn't, then what is at the bottom?
So this is actually an existential question.
Suppose we reach a final description of reality that is entirely accurate (no longer an approximation that fails to account for some structures and processes).
Won't that description consist of nothing but math?
Then, if it is complete, how does that description account for the fact that reality isn't math? If there is a difference between reality and math, can't that be described with math, so then it goes away?
If you want math to just be a description of a structure that itself isn't math, then the structure has to have attributes that are not covered by the math. The math is then using a substitute structure with fewer properties as a proxy for that structure. Once all the attributes are rolled into the math, there ceases to be a difference between that target structure and the math; it is that mathematical object. To prove otherwise requires one to name the difference. If two entities are completely described and found to be the same, they are just one entity.
> You may as well be declaring the world to be flat.
It is testable that the world isn't flat.
Example: space-time exists in n(>=4) dimensions, there are several quantum fields that exist in that space-time and evolve/interact according to the equations of qft, gravity has some mathematical description (better than the ones we have now). And that's it, there's nothing else for us to know. Of course, we can never know for certain whether we've bottomed-out our description of reality, so physicists will always have some hope.
Now, if the universe is described entirely by mathematics, bootstrapped by a few base cases (the existence of fields/particles/space-time/strings/etc), I'd be comfortable with that. I'm not sure what a truly satisfying understanding of reality would look like beyond that.
But we have that in math and logic: axioms!
> Of course, we can never know for certain whether we've bottomed-out our description of reality
That we do not know is a fact; but is it a fact that we can never know?
What if one fine day we do know?
Yes and no. Axioms are chosen such that the math that is based on them makes sense to us (because the math reflects what we'd expect in the real world) and isn't self-contradictory.
So they do exist for a reason.
> That we do not know is a fact; but is it a fact that we can never know?
What if our universe actually has randomness that appears to come from "outside the system", i.e. there is really no way to predict the outcome of certain things from inside the system.
Picture you're sitting in a box trying to play Mikado. In theory you should be able to perfectly predict the outcome of every move you make - except there's a stranger shaking the box you're sitting in at truly random intervals.
For our reality this would mean that there's no way for us to develop an understanding of certain things and predict their behavior.
So essentially the end of the line for all sciences except theology and philosophy.
This possibility makes me somewhat uncomfortable.
On the other hand maybe everything is perfectly deterministic within our universe.
> Won't that description consist of nothing but math?
Math is just words. Any description of anything can be reduced to math if you assume the proper math will be developed to describe it. But the same could be said of any subject. You could say the universe is Mandarin Chinese if you find a perfect description in that language. You could say the universe is binary code if you found a perfect description of it in binary. You could say the universe is tree bark if you found a perfect description of it in tree bark.
You're saying nothing useful. You're just confusing the map for the territory. (And arguably, you will never find a complete description of the entire universe that fits in a network of human brains, so the universe can't be math for that reason alone.)
I am not at all! We do math with symbols and diagrams, but those symbols and diagrams are about something: they are maps, and there is a territory that they are about. When I say "math", I include the concepts, not just the symbolic, graphical and other representations.
So what I'm saying is that if/when mathematics provides a complete map to that territory which is "the world", that territory must then be made of exactly the same stuff as other described-by-math territory-entities like "ellipse", "pi", "finite field", "Hilbert space", ... the map of mathematics doesn't describe any other kind of territory.
I'm not saying that the symbols and pictures used to carry out math (the "maps") are the world.
> you will never find a complete description of the entire universe that fits in a network of human brains
The description of the state of it as it is before us is monstrously large, but the initial conditions and rules might not be. E.g. the Mandelbrot set is just iterating on z^2 + c for various c. To describe isn't necessarily to evaluate.
>if/when mathematics provides a complete map to that territory which is "the world",
Then it will encode all the information that exists in the universe. Which will not fit in your head, and thus will not be mathematics. Just because a territory is trivially a map of itself doesn't mean it is useful to call it a map.
>The description of the state of it as it is before us is monstrously large, but the initial conditions and rules might not be.
Reality isn't just the initial rules. It has state. That state must be described for you to have a complete description of reality. Yes, we could mathematically describe the complete laws of nature. But reality is necessarily more than just that, because the state is really damn relevant. You can't just ignore it because it doesn't fit your argument.
If reality is deterministic, then all the state is derivable from the rules and the initial state, and it's possible we could figure out the rules and initial state and become relatively sure of them. (We probably wouldn't be able to calculate the future, since evaluating the rules within reality probably necessarily takes more steps than reality takes during that time. But we might be able to calculate enough to see that it matches up with what we think the early universe must have looked like. We might notice that even the smallest of tweaks to the rules or initial state creates an avalanche of changes that doesn't line up with what we believe the early universe looked like, so we can be sure there are no free parameters.) It's fully possible that the rules and initial state could fit in someone's head. They might be ridiculously simple, like Conway's game of life in a few more dimensions.
(If you think "reality can't be deterministic because true randomness or quantum events exist", well you can fit true randomness into a deterministic theory by having the world branch of every random event outcome. The Many-Worlds Interpretation could be viewed as doing something like this for quantum events; it's a deterministic theory.)
http://blog.rongarret.info/2015/02/31-flavors-of-ontology.ht...
> all nouns are abstract because there’s no such thing as a chair divorced from reality,
> there’s nobly instantiations of these abstract concepts we consider close enough to be
> concrete
Numbers don't exist. Take the number two. You can have two apples but that's not the number two. You can write the numeral "2", but that's not the number two. It's one line. The word "two" has three letters, it's obviously not the number two. In fact, the number two doesn't exist (or it's existence is not contingent on an arrangement of matter/energy. No pattern of matter/energy/space/time is actually THE number two.)
To me, it's wonderful to reflect on the epistemological status of numbers. They don't seem to exist, but we talk about them as if they do. Certainly they are useful. But they don't exist any more than, say, Sherlock Holmes does.
It's also much fun to reflect that, whatever the epistemological status of number, that status is shared by computer languages and algorithms and much else. E.g. the C language doesn't exist. The C standard(s) exist, many C compilers exist, lots and lots of C code exist, etc., but the language itself doesn't exist anywhere, any more than does the number two.
Only for the sense of "exist" which is roughly equivalent to "can poke a hypothetical indestructible stick at it, given enough energy."
the C language doesn't exist. The C standard(s) exist, many C compilers exist, lots and lots of C code exist, etc., but the language itself doesn't exist anywhere, any more than does the number two.
For the purpose of tax law, software is defined as a "tangible good." However, in other areas of the law, it's considered intangible.
There is the Tao and then there are the 10,000 things.
If you buy the above, you’d have to say that a single apple is also “one”. After all, it’s just another broad category that a single apple fits in to. If not, I think you’d have to reject the fact that a honeycrisp is an apple, which seems untenable.
Even if I agree that THE number "one" is "just another broad category that a single apple fits in to", I think the epistemological status of the Broad Category is then itself just as problematical, existence-wise, as the original concept of "number", eh? We can at least display a pair of apples and "count" to "two" with them, etc. What hope do we have of enumerating (no pun intended) the Broad Category of "one"? And what do we gain from introducing it?
The experience of numbers certainly exists. The reason you recognise one-ness and two-ness in objects is because your brain experiences a difference between one-object and a-pair-of-those-objects.
The experience is subjective. If you have two objects that are identical except for the fact that one is red and the other is blue, you can parse them as one-each-of-red-and-blue or as two-identical-things. Both interpretations are logically consistent, but the most useful interpretation depends on the context, and your specific needs as defined by the context.
It's easy to underestimate how subjective math is.
I've long suspected that this is why math eventually dissolves into incompleteness theorems. You can't prove a subjective experience, and eventually all mathematical reasoning reduces to subjectivity - even something basic like true/false, which is essentially just an interpretation of experience. (And can be very unreliable.)
Math is really an introspective map of experiences and processes that we find collectively consistent. We like to tell ourselves it's external and objective. But how can we tell the difference between true objectivity and subjective consistency when we only have shared human subjectivity to use as a reference?
I assume you find the above deeply unsatisfying. Most people do. Yet I find it clarifies the matter entirely! If a person's conception of "number" is consistent with the Peano axioms, and if that person's use of the word "exists" includes the existential logical quantifier, then for that person "numbers exist" is a theorem. If a person uses other definitions, the matter may be otherwise.
[1]: https://plato.stanford.edu/entries/logic-classical/ [2]: https://en.wikipedia.org/wiki/Peano_axioms [3]: https://en.wikipedia.org/wiki/Existential_generalization
Also interesting to try to show "zero" of something. Zero is fun because it took longer to exist as a number than one & two, and there was some debate over whether it should be a number. "How can nothing be something?"
My abstract algebra professor would be horrified by my misstatement, and I shall go to bring flowers as a sign of atonement and rememberance to his grave at the earliest available opportunity.
Maybe my example wasn't the best but this “democracy thing” gets discussed as if it has material properties: that it can expand, that it can be transported from one place to another, that it can be expelled from a locale, and so forth. Those are pretty material attributes to assign to something that is actually notional.
I think you're caviling, at this point.
Look, we could go on with this all day. What’s your point? To show your terminology and grasp of topics is greater than mine? Is this some kind of passing match? Because I’m really not interested in these schoolyard antics.
So, we can create programs that appear to do intelligent things, but it is the human observer that brings meaning to what the program is doing. The program might output the letters "dog" when a dog pass in front of its camera, but its the human that sees the "dog" that knows what a dog actually is.
I get the feeling you’re not quite so well versed on this stuff as you think you are. Want a true example of ‘one’: the number of solutions to the intersection of two linear equations that are not parallel. Want an example of ‘two’: the number of solutions between a quadratic equation and a linear equation (given that certain existence criteria are met).
I think you need to revise your measure theory.
Maybe you're reading too much into this.
As for reading “too much into this”, as Instated: Irma a mathematician, of course I’m going to take this seriously and make an effort to disprove naive misconceptions.
- Child is put into perfectly sealed cube. They have a torch, which provides light. Since the light cannot escape the cube, what happens when they turn the torch off? Where does it go?
- If you put a brain in a jar and wired up electrodes which pulsed up every nerve ending etc to simulate reality - would the brain know it was in a jar? How? (The matrix being the modern representation)
-
Observed externally the mass of the cube (assuming you didn’t provide a life-support system as I suggested) would remain unchanged because all mass energy is still within the cube. It has simply rearranged itself in a manner unobservable to an external observer.
When they turn the light off you have a black body cavity with an intruder. Ultimately all the light would get reflected back and forth until it fell on various parts of his body whose colour allows them to absorb the incident light waves. The child would get slightly, probably imperceptibly warmer. His surface molecules would move faster and thus he’d have more heat energy.
It was a simply worded but delightfully insightful wondering. I did my best in the moment and answered, "One way we can make numbers is by counting." I've continued to mull this question over since then. I'm drawn to this child's conception of numbers as something we make. It makes sense to me as a teacher of young children and a wonderer about math myself, yet it doesn't really square with what we've been told about numbers in most educational settings.
- Differences of counting numbers (integers)
- Ratios of numbers (rationals)
- Limits to sequences of numbers (reals)
- Solutions to polynomials made from numbers (complex)
You can think of each of these as adding a layer of behavior to the basic “each number is nothing or one more than another number”.
The second to last, that should be Cauchy sequences.
And for the last, demonstrating that the algebraic closure of the reals is the complex numbers from first principles is much harder than describing complex numbers as pairs of reals with a multiplication rule that (a,b) * (c,d) = (ac - bd, ad + bc) and then much later proving that it is algebraically closed through complex analysis.
For those who don't know the proof, the idea is this. Liouville's theorem says that if a function is differentiable everywhere, and it is bounded, then it must be constant. Now suppose that p(z) is a polynomial. Consider the function 1/p(z). You can show that as z approaches infinity, it approaches zero. It is not constant. Therefore it must not be differentiable or not bounded. It doesn't take too much work from there to prove that it blows up somewhere, and the spot that it blows up is a point where p(z) is 0.
Apply unique factorization for polynomials (see http://sites.millersville.edu/bikenaga/abstract-algebra-2/po... for that proof) and you quickly get the fact that the complex numbers are algebraically closed.
It sounds harder, but is actually easier to go through the Cauchy sequence definition and then point out that the decimal representation naturally gives rise to a Cauchy sequence. So, for example, 3.1415926535... gives you (3, 31/10, 314/100, 3141/1000, ...). And as Cauchy sequences, of course, (1, 1, 1, 1,...) is easily proved to be the same as (9/10, 99/100, 999/1000, ...).
The equivalence relationship is that the sequence (x_1, x_2, x_3, ...) is equivalent to (y_1, y_2, y_2, ...) if and only if the limit as n goes to infinity of x_n - y_n = 0.
Formally, the real number represented by (x_1, x_2, x_3, ...) is the set of all Cauchy sequences which are equivalent to that one. Since "equivalent to" is transitive, any Cauchy sequence in that set will define the same set.
Addition and multiplication are defined elementwise. Proving that they are well-defined is relatively straightforward. Their algebraic properties follow for free. Any rational number q can be mapped to the Cauchy sequence (q, q, q, ...) which leads to a unique real number that we somewhat sloppily call q again.
I've left some details out, but this construction is well-understood, and is how we define the completion of a metric space.
There are whole numbers (integers, terminology not accurate but it'll do for now), and then there are numbers called fractions (rationals) which are either whole numbers themselves, or they fall between two whole numbers, and we get them by dividing one whole number by another.
The real numbers are either fractions, or they are numbers that fall between two fractions. If it's one inch from the center of a circle to its edge, the distance around the circle is not a fraction. In inches it's about 3.14; it falls between 3 and 3 1/7, and no fraction you can possibly choose will ever get it exactly, it will be at best a little under or a little over.
You can get other non-fraction real numbers by finding square roots, but there are infinitely many of these numbers between any two fractions, not all of which are square roots. And if you try to take the square root of a negative number, you won't get a real number at all, and we call these numbers "imaginary".
Hearing stuff like this as a kid blew my mind, a bit like learning about black holes. It set me up to enjoy math throughout my life.
Was there ever a proposed type of number that is today not considered a number? Like i=squareroot(-1), was there ever an attempt to do math with something like y=1/0 or another illegal construct?
Wheels are a type of algebra where division by 0 is defined.
https://en.wikipedia.org/wiki/Wheel_theory
https://math.stackexchange.com/questions/994508/wheel-theory...
To call wheels "obscure" is to make them out to be better-known than they are, however.
But eek! From the wikipedia article:
0x ≠ 0 in the general case
x − x ≠ 0 in the general case
Are there any practical uses for this kind of algebra?
If you keep factors like this, then you could implement L'Hopital's rule without the result only being true if considered under a limit: say, lim(x->0) (5x^2/ 3x^2) = 5/3 could be computed as (5)0^2 / (3)0^2 = 5/3.
This is not like Wheels though; it requires that any power of 0 be a distinct number. I of course have no idea if it is sound or meaningful, but I do find myself thinking about it a lot.
Hamilton attempted a theory of triplets before setting on 4d quaternions, so that would be one example of a failed number system off the top of my head.
Leibniz's (re?)invention of Calculus used infinitesmals. Infinitesmals as understood by mathematicians then do not correspond to any numbers we use today. (Yes, yes, something else called infinitesmals do show up in nonstandard analysis and the notation deliberately looks the same. But the underlying concepts are more..complicated.)
Can you expand on this?
Yes, maybe more than you want. :-)
Abraham Robinson's version of non-standard analysis went like this.
You start with the standard model of the real numbers and associated concepts like sets, functions, and so on. Using something called the ultrafilter construction, you construct a new, larger model of the real numbers+stuff, and a mapping from the standard model to the nonstandard model. So we have nonstandard numbers, nonstandard sets, nonstandard functions and so on. Some of which are just mappings of the standard ones, and others of which are new.
Thanks to something called the transfer principle, all statements in first order logic about things involving real numbers remain true about the nonstandard versions of the same.
In other words within the nonstandard model, the world looks the same as within the standard model. But there is a key feature. From the construction we know that the nonstandard model has numbers in it that are closer to 0 than any real number except 0. That set is the infinitesmals. The infinitesmals are a set identifiable from the construction, but are NOT a nonstandard set. And 1/infinitesmal gives you infinite numbers whose absolute value is larger than any standard real.
Now here is the point of the whole construction. Suppose that (f(x + dx) - f(x))/dx only varies by an infinitesmal from a non-standard version of a real across all possible infinitesmals. Then it turns out that that real is the derivative in the usual notation. Similarly if a Riemann sum that breaks an area into N pieces always gives the same answer to within an infinitesmal for all infinite integers N, then it turns out that the function is Riemann integral, and that answer is the actual integral. And with these two insights, most of the handwavy arguments that people used to use can be rescued.
Some mathematicians have found that the infinitesmal notation helps them think through problems, and a some important new theorems were proven using this approach. However those proofs can be translated back to more standard notation. Many students intuitively prefer the infinitesmal notation over limits. But when you unpack it, does it make sense to invoke the axiom of choice to define the derivative? (The ultrafilter construction uses the axiom of choice. Alternatives exist, but they use complex model theoretic mechanics under the hood. Really understanding them requires a lot of machinery.)
[1] https://en.wikipedia.org/wiki/Non-standard_analysis
[2] https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis
One example of previous attempts which we'd now consider to be invalid is a lot of operations on infinite series. In early days of analysis you'd get people concluding things like "1 - 1 + 1 - 1 + ... = 1/2", which gets you into more hot water the more you look at it. The problem here isn't that they are philosophically unsound per se - you can define all sorts of notions of "infinite sum" that make this work, like cesaro summation - but they don't behave as nicely as people intuitively expected them to and the naive versions of them don't really work.
PS: this article/video essentially defines the naturals from the fundamental set theory axioms. This other video (https://www.youtube.com/watch?v=KTUVdXI2vng) from the same PBS series shows how you can then use the naturals to construct other types of common numbers, up to the reals.
>To discuss this subject we start in the middle. We suppose that we already know what integers are, what zero is, and what it means to increase a number by one unit. You may say, “That is not in the middle!” But it is the middle from a mathematical standpoint, because we could go even further back and describe the theory of sets in order to derive some of these properties of integers. But we are not going in that direction, the direction of mathematical philosophy and mathematical logic, but rather in the other direction, from the assumption that we know what integers are and we know how to count. ...
For valid starting points, that is true, but it doesn’t follow from the text. The loop has an error exit (”If the sequence starts with − then something has gone wrong”), and the text doesn’t show that you won’t get there for valid inputs, and showing that isn’t trivial.
For example, if the sequence is ++---+, the first iteration removes the second and third item, leaving +--+, and the second iteration removes the first and second item, yielding -+.
The ‘program’ crashes there because the input is invalid, but proving that it never will crash for valid inputs without resorting to “that’s how integers behave” isn’t trivial.
In case anyone just wants actual answers without reading pages and pages of prose, here is one set of constructions (there are other competitors too).
The natural 0 is the emptyset. The natural 1 is the singleton {0}. The natural 2 is {0,1}. In general, the natural n+1 is {0,...,n}. Exercise to the reader: define appropriate arithmetical functions on the naturals.
Define a relation ~ on pairs (a,b) of naturals by saying (a,b)~(c,d) if and only if b+c=a+d. Exercise to the reader: ~ is an equivalence relation. Its equivalence classes are called "integers". The equivalence class containing (a,b) represents the integer b-a. For example, the integer -1 is the equivalence class {(1,0),(2,1),(3,2),...}. Exercise: define appropriate arithmetical functions on the integers.
Define a relation ~ on pairs (m,n) (n nonzero) of integers by saying (m,n)~(p,q) if and only if mq=pn. Exercise: Show ~ is an equivalence relation. Its equivalence classes are called "rationals". The equivalence class containing (m,n) represents the rational m/n. For example, 1/2 is the equivalence class {(1,2),(-1,-2),(2,4),(-2,-4),...}={(k,2k) for all nonzero integers k}. Exercise: define arithmetic operations on the rationals.
To get from the rationals to the reals, see https://en.wikipedia.org/wiki/Dedekind_cut
Other axioms of set theory are used to formalize other steps in my post. https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... For example, the Axiom of Infinity is used to formalize the handwavy part where I said "In general, the natural n+1 is {0,...,n}".
At the end of the day, all of our Mathematics rests on foundations that are made up.
It's difficult to say what the empty set is. Because it isn't really anything at all.
Are you trying to argue coming up with new definitions isn't worthwhile if the axioms don't have a foundation in reality? If so, why? If not, what are you saying?
What I'm saying is: questioning foundations leads you to new foundations. New foundations that you can then question all over again. It's turtles all the way down.
If you're actually looking to use math, this is a futile exercise. It works, so just use it. Essentially, I am re-iterating von Neumann's statement:
> In mathematics you don't understand things. You just get used to them.
Either there is an empty set, or there isn't. If there is one, I win. If not, then let S be the set of all empty sets. There are none, so S is the empty set and I win again.
The original post is called "What even is a number?". The comment I replied to tried to answer that question with "some formalism".
What I'm saying is: questioning foundations leads you to new foundations. New foundations that you can then question all over again. It's turtles all the way down.
If you're actually looking to use math, this is a futile exercise. It works, so just use it. Essentially, I am re-iterating von Neumann's statement:
> In mathematics you don't understand things. You just get used to them.
Yep, one of my favorite math quotes and highly under-rated.
One thing I've found is that if you find yourself using the wrong "types", you're almost certainly going in the wrong direction. For example, if you're trying to solve a complex analysis exercise and you find yourself thinking about how the complex numbers = the real-coefficient polynomials modulo the ideal (x^2+1), STOP: You're getting nowhere! :)
Did you have some further point?
If you were hoping that mathematics would have - or thinking that it _should_ have - some ontological foundation that is not in this sense "made up", I'm afraid it simply doesn't.
Replies that consist solely of throwing a quote at someone are kind of rude even if you're in the right.
Responding to someone's definition that natural 0 = {} that mathematics is a "castle built on sand" isn't exactly a cogent criticism of set theory.
Auto-correct at work, I guess. For those not familiar with the term: if Google says did you mean equivalence relation, Google is right.
For anyone who wants to learn more about this: what the parent comment is describing is the construction of natural numbers as cardinalities of sets. This is the (modified) Frege definition which avoids Russell's paradox - you can use that as a jumping off point for further dives into set theory and number systems.
To define Peano arithmetic you start with a successor function, then an addition function, then multiplication, etc.
Type theory and category theory give us a much better way of constructing these sorts of objects without having to resort to creating constructions with all sorts of side effects. Not to mention that this is one possible construction of the natural numbers in set theory; in other formulations the theorem "is 2 a member of 4" would not be true, making things even more confusing. For example, we have the other standard approach of 0 = {}, 1 = {{}, 0}, n={{}, n-1}. The untyped lambda calculus and things like Church numerals have the same defect.
If we can start assigning types, then we can reason about these things in a much more concrete manner without these nasty side effects of theorems that only make sense because of definitional shortcuts. We can even derive isometries between different derivations of the same objects to show that two different definitions (like the Peano numerals and binary numbers) of a "natural number" can be used exactly interchangeably, because theorems like "is 2 a member of 4" are simply not expressible given the derivation.
Why? Any set of 4 things also contains 2 things. Where's the nonsense? Within this definition, that's consistent.
> Type theory and category theory give us a much better way of constructing these sorts of objects without having to resort to creating constructions with all sorts of side effects.
Out of curiosity, what is the category theoretic construction that avoids Russell's paradox? I don't know that there isn't one, but I can't think of it off the top of my head. I know there are category theoretic constructions in general (I responded to one someone else posted in this thread).
Talking about concepts like "the category of categories that don't contain themselves" is kind of navel-gazey; and ends up falling apart in most constructive variants just because you can't give a comprehensive construction of elements of this category.
Admittedly I'm throwing some concepts from intuitionalism and type theory in the mix here; if I took the time I could make these statements more precise.
Alternatively, what are you selecting as the objects for your category if not sets of a given cardinality?
To be clear, I consider a lot of discussion about the foundations of set theory (and paradoxes thereof) to be pretty navel-gazey.
That's the problem -- it's only within this definition. In other definitions, notable the nesting example that I gave, that theorem is false. And the theorem makes no sense in and of itself, because we're talking about numbers, so it is unexpected that the membership operator would apply at all.
Whereas I can take two definitions of the natural numbers
PN =
Peano_One: PN
Peano_Succ: PN -> PN
BN =
Binary_One: BN
Binary_2x: BN -> BN
Binary_2xp1: BN -> BN
Stated another way, consistency is only a coherent mathematical concept from the perspective of a specific set of definitions. There's no problem here.
If you give me a statement saying "For two sets, A and B, is 'A union intersection B empty", the statement would not be well-formed, so you could just say that it is not a valid proposition without asserting anything about its truthiness. Similarly, a good foundation of mathematics should be able to reject a statement like "2 is a member of 4" as poorly formed, and make no statement about its truthiness.
I guess what I'm saying is that set theory is a bad basis for mathematics and should just be regarded as an interesting relic from previous attempts to formalize mathematics so that we can move on to more powerful systems.
Counts of things are described here, but the greeks went to great lengths with geometry to discover numbers (and even killed people who came up with irrational answers)
Still, lovely article.
It's almost a dead giveaway for excerpts like 'The notion of a counter is not that of a single concrete class of object, it is an abstract description of the behaviour of a system providing certain operations satisfying certain rules.' , and 'A number is not any one thing, it is any one of any number of things that implement some operations (and there are different types of number depending on what operations you want to implement).'
My actual degree, which is where most of my philosophy of mathematics opinions were developed, is in very pure mathematics. I've done quite a lot of software development since and that's definitely shaped the framing, but the core philosophy is one that I've had since long before I knew much about computer science at all.
It's a category. 1 is the category of singletons, 2 is the category of pairs, etc.
As to what numbers are, they're an ordered collection of categories.
Edit: To whoever downvoted this comment, kindly explain. Insofar as I'm aware this is textbook maths, psychology, and philosophy.
But if you need to avoid Russell's paradox the sets defining each natural n can't contain n as an element. The easiest such construction was outlined elsewhere in this thread by xamael. Under your category theoretic construction, every natural number n is defined as the category of sets having cardinality n. But then every nth category will necessarily contain infinitely many sets with cardinality n that also contain n. That causes the paradox.
Unfortunately I don't think you can construct the naturals in a category theoretic way while avoiding Russell's paradox since any category by cardinality will fall into that trap. But if you don't need to mind that problem, this is neat.
Responding to your edit: I didn't downvote you; in fact I upvoted this comment because it's correct and it was gray at the time of my writing. My comment is just a point of clarification.
If you define the natural number 2 to be the category of pairs, your objects are the sets with cardinality 2, and your relations between objects are equivalence relations. As a consequence your category 2 will contain sets which contain itself.
https://www.math.toronto.edu/mathnet/questionCorner/epii.htm...
The computer your are using would not work without sine waves. The nanometer scale chips couldn't be manufactured without them. The high speed signals transmitted over its wiring would not reach their destination without knowing how to manipulate them.
A deeper question is what to make of "uncountable" transcendental numbers.
A number is a sequence of 1 or more digits, optionally beginning with a '-' and optionally followed by a '.' and one or more digits, where digits are the characters {'0' '1' '2' '3' '4' '5' '6' '7' '8' '9'}.
Examples: 1, 25, 3.2, -8
"1 + 2" is not a number, but its evaluation is
That is for +integer.
Not sure about how natural is real number other than pi using geometry. And irrational also use geometry.
Use algebra and group-ring-field is very late.
"1" is the set of all things which have exactly one element. It is an abstraction for all such things.
What does it mean to "have one element"? I think that's a more difficult question. Maybe it should be an axiom?
So a number can be seen as a metaphysical representation of energy.