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I had a fun mind-play with my kids; I asked them if numbers like "one" and "two" really exist. They said yes, of course. OK, I said, show me a plain old "one".
by cobbzilla 8y ago
I had a fun mind-play with my kids; I asked them if numbers like "one" and "two" really exist. They said yes, of course.
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.htm https://www.sciencedaily.com/releases/2008/07/080714111940.h...
- mzs 8y agohttps://books.google.com/books?id=1s5EAAAAIAAJ&pg=PA9&ci=137%2C231%2C774%2C616&source=bookclip https://books.google.com/books?id=1s5EAAAAIAAJ&pg=PA9&ci=137...
- deleted 8y ago[deleted]
- mjburgess 8y agoThat'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 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
- cobbzilla 8y agoYes, I could have really tortured my kids and said, you can't even show me a "ball", because it will always be a specific ball and not the archetypal concept of "ball". I probably should, it'd be fun and they're a bit older than when I pulled the "one" game.
- kazinator 8y agoI 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.
- stcredzero 8y agoI believe that in fact the numbers do exist as physical objects exist. Just in their own universe. 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.)
- kazinator 8y ago> How is it any different than believing that unicorns exist as physical objects, just in their own universe? 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.
- stcredzero 8y agoWe 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". 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/according_to_my_dodgy_backofthecoaster_maths/ https://www.reddit.com/r/mylittlepony/comments/mrs81/accordi... Do you have a good definition of what constitutes a "well-developed" universe?
- devilshaircut 8y agoAgree. Maybe it is more helpful to think of numbers as adjectives than nouns. Oneness plus twoness equals threeness. The part of speech is sort of irrelevant but it obviates a debate over whether or not instantiation of the unattached idea of a number is important, while still allowing math to occur.
- lisper 8y agoThe "technical" term amongst philosophers is "ontological category": http://blog.rongarret.info/2015/02/31-flavors-of-ontology.html http://blog.rongarret.info/2015/02/31-flavors-of-ontology.ht...
- qubex 8y agoThis reminds me of my Italian grammar teacher telling me that nouns can be broken into two categories: ‘concrete’ and ‘abstract’ (for example, ‘chair’ and ‘love’ respectively). Then came a totally mind bending digression for my eight-year-old mind when she mumbled “well actually 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”. That totally blew my mind. Together with reading The Computational Beauty Of Nature” when it first came out nine years ago those were the two key experiences that drove me towards choosing to get a degree in mathematics.
- mcv 8y ago> 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 Very Plato for an 8 year old.
- analreceiver 8y agohttps://en.wikipedia.org/wiki/Prat%C4%ABtyasamutp%C4%81da https://en.wikipedia.org/wiki/Prat%C4%ABtyasamutp%C4%81da
- carapace 8y agoAh! I wrote the below before seeing your comment: 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.
- stcredzero 8y agoNumbers don't exist. 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.
- jessermeyer 8y agoNumbers do not have a physical existence, but we can talk about them because in some capacity they absolutely do exist: conceptually. There is the Tao and then there are the 10,000 things.
- johnfn 8y agoDoes an apple exist? You can show me an apple, but I would wag my finger in exactly the same way you are to people trying to show you “one”. The apple you’re showing me is not the same thing as the word “apple” which is an abstract category that encompasses all types of apples - honeycrisp, Granny Smith, etc. You could say a honeycrisp is like an instance of apple which is a base class, so then it is indeed an apple. But then youd have to say an apple is also a “fruit” and also a “food” even though those words also encompass broad categories of 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.
- sound1 8y agoBy extension, does God exist? I don't know the answer but he seems to be very useful (at least to some members of the society!)
- r00fus 8y agoSure (s)he does - inside all of us. This works for both spiritual as well as a human nature points of view.
- dahart 8y ago> 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?"
- mzs 8y agoHistorically 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.
- qubex 8y agoHate to be pedantic, but ℤ︎ is a set (not a group) and even with 0 adjoined it only forms a group under the operators of addition, subtraction, and multiplication.. If you want a set that also forms a group under division you need to ‘upgrade’ to the set ℚ︎.
- mzs 8y agoℤ︎ (…, -2, -1, 0, 1, 2, …) and + forms a group. It is also a ring with * and +. ℚ︎ with * and + forms a field.
- hopler 8y agoYes, that's what qubex said, in correcting the error in your previous comment.
- mzs 8y agoℤ︎ with * is not a group, there is no inverse for 0. When being pedantic it's good to be correct.
- qubex 8y agoYou are perfectly correct, I was wrong. Upon closer reflection it appears impossible to have a nontrivial set (meaning other than {0}) that forms a group both under addition and under multiplication because the identity element of the former does not have an inverse in the latter. However, take this with a grain of salt, because I haven't had time to prove this to my own satisfaction yet (and I am loathe to go googling for answers in my own field of expertise). 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.
- snarfy 8y agoYou might even say the unattached "one" is more real than the fork. At the end of the universe, that fork is long gone, but "one" is still there.
- Nomentatus 8y ago"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.
- qubex 8y agoI’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 the overarching ideal.
- Nomentatus 8y agoThat's not what reification means, it's more a form of Platonism (Idealism), plus. See The Meno and it's discussion of the forms (say, the "form" of a spoon.) Plato would reify democracy (although he hated it) but most of us certainly do not. A nominalist (even a Berkeley follower) could discuss democracy perfectly well.
- qubex 8y agoI'm pretty well versed in what reification means. Fundamentally it entails taking a theoretical concept and treating it as if it were a tangible, almost embodied, thing. 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.
- Nomentatus 8y agoNow if you look up tangible... I think you're caviling, at this point.
- kirkules 8y agoYou should keep in mind that being unable to show or find an example doesn't disprove the category. You'll never show me a fork yesterday, but that doesn't mean there were no forks yesterday.
- cicero 8y agoThis is why Aristotle said that the intellect cannot be material. The human intellect is able to receive and understand abstractions. However, whenever you try to represent an abstraction as a physical object or in a physical medium, it is no longer abstract. It is true that you can write about abstractions using physical ink on physical paper, but those are just symbols; they are not the abstraction. It is only when a person reads the symbols, understands them, and forms the abstraction in his mind that the abstraction comes into existence. 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.
- hopler 8y agoThat sounds like the illusion of consciousness. Neural networks show that you don't need "dog" in order to do whatever you need to with the idea of a dog.
- qubex 8y agoI’m a mathematician and you seem a bit confused. Natural numbers ({1, 2, 3, ... }) are for counting. Giving a bunch of objects as being prime examples of that which can be counted with natural numbers is as close as one can get (short of just scribbling down the numerals and kind-of circumventing the point). Other classes of numbers (particularly the Reals) are for measuring. Again, short of writing down examples of decimal expansions, a child could point to a (non-integer) mark on a measuring tape. 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.
- WilliamEdward 8y agoI think they're just trying to say "1" is not a tangible object in the physical universe, not that it does not exist as a concept. The natural numbers are just data, information, they don't exist beyond the first dimension except as descriptors to things in further dimensions, just like lines can be built up into higher dimensions but cannot be physically created for us to observe. Maybe you're reading too much into this.
- qubex 8y agoHow is two straight lines crossing in one and only one place not an example of a single occurrence without reference to real world objects (which the OP dismissed as “not really being a number”). 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.
- samhain 8y agoSo by your definition, one is an adjective, so asking them for a noun that represents the adjective "one" violates the definition. It's sort of a trivially false disingenuous question.
- Ntrails 8y agoA coiuple of my favourites: - 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) -
- qubex 8y agoIn the case of the cube the internal temperature would rise slightly. Th kid is also emitting some heat too so that would also contribute to the rising temperature. Eventually the kid would die from lack of oxygen. If you provide oxygen to the child and remove carbon dioxide the system would no longer be sealed: adiabatic expansion of added oxygen would lower the temperature and removal of carbon dioxide would extract some heat from the cube. 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.