131–140 of 173 posts
This really just boils down to: Counting is an intuitive thing for us and it's useful to mathematically define it.
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One of the Peano axioms is that every natural number is either zero or the successor of another. It's easy to explain to kids without jargons: a number either counts nothing at all, or it counts things that have an extra item compared to some other thing.
This kind of breaks down once you move past natural numbers, but for early learners of mathematics i agree it's useful to think about.
- 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”.
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Don't even numbers have a pretty easy definition? If prime factorizartion of the number contains at least a 2
Oh sure, but then the article started talking about sheep and rocks and I'm like "Oh boy, I bet I totally don't understand what it means to be even !"
It's extremely obvious that the author is a computer scientist
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.
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> Unicorns. Kirin. Narwhals. Rhinos. So you have two independent origins in culture for such a creature, plus two appearances of something similar in nature. 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…
Kirin is giraffe and upset Confucius as it should not appear in his chaotic time.
> You can see some of that history of what we call some of the different sorts of numbers: e.g. Negative, irrational, and imaginary numbers. Each of these represents a bitter argument about whether a new type of number should be allowed, all of which were eventually won by the people on the side of the new numbers... Was there ever a proposed type of number that is today not considered a number? Like i=squareroot(-1)…
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.
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> You cannot show just "one" unattached to anything else. 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?"
Historically true, but glad it's there so ℤ is a group. The way to think about 0 for a child is what you add to anything so it stays the same, ie nothing.
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This kind of breaks down once you move past natural numbers, but for early learners of mathematics i agree it's useful to think about.
But other kinds of numbers can be made from that: - 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”.
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"Reification" is the term to search for if you wish to delve into the extensive Philosophical bibliography on this topic that's accumulated over the last few centuries. Within mathematics "Intuitionist Mathematics" is the rubric for the anti-reification view. This relies on computability as a substitute for "existence." The Intuitionist view has pretty much triumphed.
I’m all for reification. It’s not even limited to mathematics: ‘democracy’ is just another example of a reification from another realm of human endeavor: it gets implemented in different countries with slightly different axioms and varying degrees of success but in aggregate those nations that abide by some form of it are accepted as functional equals by other whom have likewise implemented their own instantiation of…
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That's just "Exist as physical objects exist" 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 no…
I believe that in fact the numbers do exist as physical objects exist. Just in their own universe. 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.