The most commonly used/accepted foundation for mathematics is set theory, specifically ZFC. Relations are modeled as sets [of pairs, which are in turn modeled as sets].
A logician / formalist would argue that mathematics is principally (entirely?) about proving derivations from axioms - theorems. A game of logic with finite strings of symbols drawn from a finite alphabet.
An intuitionist might argue that there is something more behind this, and we are describing some deeper truth with this symbolic logic.
Vast piles of mathematics exist without any relational objects, and not exclusively in the intuitionistic sense either. Geometers say it's about rigidity. Number theorists say it's about generative rules. To a type-theorist, it's all about injective maps (with their usual sense of creating new synonyms for everything).
The only thing these have in common is that they are properties about other properties.
I prefer a more direct formulation of what mathematics is, rather than what it is about.
In that case, mathematics is a demonstration of what is apparent, up to but not including what is directly observable.
This separates it from historical record, which concerns itself with what apparently must have been observed. And it from literal record, since an image of a bird is a direct reproduction of its colors and form.
This separates it from art, which (over-generalizing here) demonstrates what is not apparent. Mathematics is direct; art is indirect.
While science is direct, it operates by a different method. In science, one proposes a hypothesis, compares against observation, and only then determines its worth. Mathematics, on the contrary, is self-contained. The demonstration is the entire point.
3 + 3 = 6 is nothing more than a symbolic demonstration of an apparent principle. And so is the fundamental theorem of calculus, when taken in its relevant context.
To form or even to define a relation you need some sort of entity to have a relation with.
My wife would have probably gone postal (angry-mad) if I had tried to form an improper relationship with her. It turns out that I needed a concept of woman, girlfriend and man, boyfriend and then navigate the complexities involved to invoke a wedding to turn the dis-joint sets of {woman} and {man} to form the set of {married couple}. It also turns out that a ring can invoke a wedding on its own but in many cases, it also requires way more complexity.
You might start off with much a simpler case, with an entity called a number. How you define that thing is up to you.
I might hazard that maths is about entities and relationships. If you don't have have a notion of "thingie" you can't make it "relate" to another "thingie"
It's turtles all the way down and cows are spherical.
"Mathematics is a field of study that discovers and organizes methods, theories, and theorems that are developed and proved for the needs of empirical sciences and mathematics itself."
In order to understand mathematics you must first understand mathematics.
> I think the biggest mistake people make when thinking about mathematics is that it is fundamentally about numbers. It’s not. Mathematics is fundamentally about relations.
Eh, but you can also say that about philosophy, or art, or really, anything.
What sets mathematics apart is the application of certain analytical methods to these relations, and that these methods essentially allow us to rigorously measure relationships and express them in algebraic terms. "Numbers" (finite fields, complex planes, etc) are absolutely fundamental to the practice of mathematics.
For a work claiming to do mathematics without numbers, this paper uses numbers quite a bit.
Interesting paper; had not known of this earlier. Thanks for posting.
Mathematics is the study of Abstractions and Modeling using these abstractions. Entities/Attributes/Rules establishing Relationships (numerical and otherwise) all fall out of this.
The best way to understand this is through the idea of a Formal System - https://en.wikipedia.org/wiki/Formal_system All that the common man thinks of as "Mathematics" are formal systems.
Because HN readers can't know if the summary is an accurate representation of the original article, nor what detail or nuance has been winnowed out in the summarizing process. But if there is a summary that seems "good enough" to form an opinion, then the discussion on HN will be based on the summary, not on the complete article. We see the same thing with editorialized titles.
A better way to get a taste of the article is to look over the HN discussion. The top comment(s) should give people a hint as to what it's about and whether it's worth the time to read the whole thing. Otherwise just reading the HN discussion should be a good way to get the jist of it. But that only works if enough of the commenters have actually read the whole article rather than a summary.
Aren’t many algebraic results dependent on counting/divisibility/primality etc...?
Numbers are such a fundamental structure. I disagree with the premise that you can do mathematics without numbers. You can do some basic formal derivations, but you can’t go very far. You can’t even do purely geometric arguments without the concept of addition.
Addition does not require numbers. It turns out, no math requires numbers. Even the math we normally use numbers for.
For instance, here is associativity defined on addition over non-numbers a and b:
a + b = b + a
What if you add a twice?
a + a + b
To do that without numbers, you just leave it there. Given associativity, you probably want to normalize (or standardize) expressions so that equal expressions end up looking identical. For instance, moving references of the same elements together, ordering different elements in a standard way (a before b):
i.e. a + b + a => a + a + b
Here I use => to mean "equal, and preferred/simplified/normalized".
Now we can easily see that (a + b + a => a + a + b) is equal to (b + a + a => a + a + b).
You can go on, and prove anything about non-numbers without numbers, even if you normally would use numbers to simplify the relations and proofs.
Numbers are just a shortcut for dealing with repetitions, by taking into account the commonality of say a + a + a, and b + b + b. But if you do non-number math with those expressions, they still work. Less efficiently than if you can unify triples with a number 3, i.e. 3a and 3b, but by definition those expressions are respectively equal (a + a + a = 3, etc.) and so still work. The answer will be the same, just more verbose.
I think the biggest mistake people make when thinking about mathematics is that it is fundamentally about numbers.
It’s not.
Mathematics is fundamentally about relations. Even numbers are just a type of relation (see Peano numbers).
It gives us a formal and well-studied way to find, describe, and reason about relation.
The most commonly used/accepted foundation for mathematics is set theory, specifically ZFC. Relations are modeled as sets [of pairs, which are in turn modeled as sets].
A logician / formalist would argue that mathematics is principally (entirely?) about proving derivations from axioms - theorems. A game of logic with finite strings of symbols drawn from a finite alphabet.
An intuitionist might argue that there is something more behind this, and we are describing some deeper truth with this symbolic logic.
Vast piles of mathematics exist without any relational objects, and not exclusively in the intuitionistic sense either. Geometers say it's about rigidity. Number theorists say it's about generative rules. To a type-theorist, it's all about injective maps (with their usual sense of creating new synonyms for everything).
The only thing these have in common is that they are properties about other properties.
You just said the same thing as GP, but it sounds like you’re trying to argue with them about it.
Perhaps there’s a math formula to describe the relation between your messages’ properties.
But that thing the property is from is called a number, isn't it?
I prefer a more direct formulation of what mathematics is, rather than what it is about.
In that case, mathematics is a demonstration of what is apparent, up to but not including what is directly observable.
This separates it from historical record, which concerns itself with what apparently must have been observed. And it from literal record, since an image of a bird is a direct reproduction of its colors and form.
This separates it from art, which (over-generalizing here) demonstrates what is not apparent. Mathematics is direct; art is indirect.
While science is direct, it operates by a different method. In science, one proposes a hypothesis, compares against observation, and only then determines its worth. Mathematics, on the contrary, is self-contained. The demonstration is the entire point.
3 + 3 = 6 is nothing more than a symbolic demonstration of an apparent principle. And so is the fundamental theorem of calculus, when taken in its relevant context.
To form or even to define a relation you need some sort of entity to have a relation with.
My wife would have probably gone postal (angry-mad) if I had tried to form an improper relationship with her. It turns out that I needed a concept of woman, girlfriend and man, boyfriend and then navigate the complexities involved to invoke a wedding to turn the dis-joint sets of {woman} and {man} to form the set of {married couple}. It also turns out that a ring can invoke a wedding on its own but in many cases, it also requires way more complexity.
You might start off with much a simpler case, with an entity called a number. How you define that thing is up to you.
I might hazard that maths is about entities and relationships. If you don't have have a notion of "thingie" you can't make it "relate" to another "thingie"
It's turtles all the way down and cows are spherical.
A former Wikipedia definition mathematics: Mathematics is the study of quantity, structure, space and change.
Current definition:
"Mathematics is a field of study that discovers and organizes methods, theories, and theorems that are developed and proved for the needs of empirical sciences and mathematics itself."
In order to understand mathematics you must first understand mathematics.
Only mathematics can define objects in a non recursive way. Human language can’t (Münchhausen Trilemma)
> I think the biggest mistake people make when thinking about mathematics is that it is fundamentally about numbers. It’s not. Mathematics is fundamentally about relations.
Eh, but you can also say that about philosophy, or art, or really, anything.
What sets mathematics apart is the application of certain analytical methods to these relations, and that these methods essentially allow us to rigorously measure relationships and express them in algebraic terms. "Numbers" (finite fields, complex planes, etc) are absolutely fundamental to the practice of mathematics.
For a work claiming to do mathematics without numbers, this paper uses numbers quite a bit.
I think of pure math as choosing a set of axioms and then proving interesting theories with them.
Prime numbers are the queens/kings of mathematics though.
Interesting paper; had not known of this earlier. Thanks for posting.
Mathematics is the study of Abstractions and Modeling using these abstractions. Entities/Attributes/Rules establishing Relationships (numerical and otherwise) all fall out of this.
The best way to understand this is through the idea of a Formal System - https://en.wikipedia.org/wiki/Formal_system All that the common man thinks of as "Mathematics" are formal systems.
A good example is this wired article How Do People Actually Catch Baseballs? - https://www.wired.com/story/how-do-people-actually-catch-bas... (archive link https://archive.is/Aarww)
> Mathematics Without Numbers
Look inside
> Numbers
The numbers are used as labels or indicators, not for their numerical values, so I think the title is still correct.
The numerical value is a label or indicator for an abstract property of physical sets of things, so I don't see how this is anything different.
Is there a source from somewhere that didn't kill Aaron Swartz? I'd rather not reward them with a click.
JSTOR settled with Swartz and did not pursue a civil lawsuit.
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Please don't do this here. Article summaries have always been eschewed on HN.
I would have liked a summary before reading.
Why is writing a summary a bad thing?
Because HN readers can't know if the summary is an accurate representation of the original article, nor what detail or nuance has been winnowed out in the summarizing process. But if there is a summary that seems "good enough" to form an opinion, then the discussion on HN will be based on the summary, not on the complete article. We see the same thing with editorialized titles.
A better way to get a taste of the article is to look over the HN discussion. The top comment(s) should give people a hint as to what it's about and whether it's worth the time to read the whole thing. Otherwise just reading the HN discussion should be a good way to get the jist of it. But that only works if enough of the commenters have actually read the whole article rather than a summary.
Kemeny is an interesting fellow. He is part of the duo responsible for the BASIC language (at Dartmouth).
I found his book "Man and the Computer" particularly prescient.
https://en.wikipedia.org/wiki/John_G._Kemeny
https://archive.org/details/mancomputer00keme
Aren’t many algebraic results dependent on counting/divisibility/primality etc...?
Numbers are such a fundamental structure. I disagree with the premise that you can do mathematics without numbers. You can do some basic formal derivations, but you can’t go very far. You can’t even do purely geometric arguments without the concept of addition.
Addition does not require numbers. It turns out, no math requires numbers. Even the math we normally use numbers for.
For instance, here is associativity defined on addition over non-numbers a and b:
a + b = b + a
What if you add a twice?
a + a + b
To do that without numbers, you just leave it there. Given associativity, you probably want to normalize (or standardize) expressions so that equal expressions end up looking identical. For instance, moving references of the same elements together, ordering different elements in a standard way (a before b):
i.e. a + b + a => a + a + b
Here I use => to mean "equal, and preferred/simplified/normalized".
Now we can easily see that (a + b + a => a + a + b) is equal to (b + a + a => a + a + b).
You can go on, and prove anything about non-numbers without numbers, even if you normally would use numbers to simplify the relations and proofs.
Numbers are just a shortcut for dealing with repetitions, by taking into account the commonality of say a + a + a, and b + b + b. But if you do non-number math with those expressions, they still work. Less efficiently than if you can unify triples with a number 3, i.e. 3a and 3b, but by definition those expressions are respectively equal (a + a + a = 3, etc.) and so still work. The answer will be the same, just more verbose.
>Numbers are just a shortcut for dealing with repetitions
An interesting explanation, I think I agree
That is not really a very deep result.
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