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Mathematics Without Numbers (1959)
- dinkelberg 11mo ago[flagged]
- drivebyhooting 11mo agoAren’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.
- Nevermark 11mo agoAddition 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.
- drivebyhooting 11mo agoThat is not really a very deep result.
- Isamu 11mo ago>Numbers are just a shortcut for dealing with repetitions An interesting explanation, I think I agree
- EvanAnderson 11mo agoKemeny 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://en.wikipedia.org/wiki/John_G._Kemeny https://archive.org/details/mancomputer00keme https://archive.org/details/mancomputer00keme
- kuipferings 11mo ago[flagged]
- tomhow 11mo agoPlease don't do this here. Article summaries have always been eschewed on HN.
- dinkelberg 11mo agoI would have liked a summary before reading. Why is writing a summary a bad thing?
- tomhow 11mo agoBecause 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.
- jbandela1 11mo agoI 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.
- hurturue 11mo agoPrime numbers are the queens/kings of mathematics though.
- somewhereoutth 11mo agoThe 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.
- gerdesj 11mo agoTo 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.
- 7373737373 11mo agoA former Wikipedia definition mathematics: Mathematics is the study of quantity, structure, space and change.
- Zambyte 11mo agoIs there a source from somewhere that didn't kill Aaron Swartz? I'd rather not reward them with a click.
- oceansky 11mo agoJSTOR settled with Swartz and did not pursue a civil lawsuit.
- rramadass 11mo agoInteresting 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 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-baseballs/ https://www.wired.com/story/how-do-people-actually-catch-bas... (archive link https://archive.is/Aarww https://archive.is/Aarww)
- zygentoma 11mo ago> Mathematics Without Numbers Look inside > Numbers
- lambdaone 11mo agoThe numbers are used as labels or indicators, not for their numerical values, so I think the title is still correct.
- 1718627440 11mo agoThe 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.