Thread: What is a 2?

I get the idea that 0 is nothing, null, void, etc. Then, we should be able to say that 1 is simply "something", right? So, then, what is 2? Is 2 "something + something?" Of course, something + something = something.

My point is, "something" is just two vague a notion of what a number is.

So, what if we just say that 2 is twice as many as one? I hope you see the obvious flaw in that kind of answer and why we must reject it.

Do we define numbers like this:
1 = x where 0<x<2 or, more precisely: .999...<x<1.0000...1
And so, every number is defined by it's relation to other numbers.
In this system, we still don't "know" what a 1 is, or any other number for that matter, but we would at least never confused a 1 for a 5, or a 3 for a 2.
Of course, the system only works if we already know a priori that 0<1<2.

In fact, it reminds me of Aquinas' method of via negativa. Where he tried to define God by all the things he is not. "God is not physical, not in time, not... etc" However, this method surrenders all hope of finding what God IS. Unless you're of the opinion that Aquinas only used the method itself as a way to come up with an idea that no one else thought of. In a way similar to poets using rhyming to come up with lines they might not have thought of otherwise.

So, with that... How about I try using via negativa as a source of inspiration. We already know that 1 is not 0, 2, 3, or any other number. We've already eliminated an infinite amount of possibilities, and that was just my first negation. And yet, I know that 1 is the only number you can divide by itself and get itself as an answer. It's the only number that equals itself no matter what power it is raised by. It's also the only value of .

So, one negation in and I've already found a plethora of attributes unique to 1. Perhaps this would be the best definition of 1, ...?

So, would it be fair for me to define the rest of the numbers based on the number 1? In other words, ? I think the answer is obvious from the moment I wrote out that equation. I can't use the concept of 2 to define 2!

Maybe I should try via negativa again? Just for old times sake...
I know that 2 is not 1, 0, 3, etc... I could say that 2 is the common factor all even numbers can be divided by, but of course, this is how we define even numbers, so it's only a more complicated circular argument.

What about this: We already understand 0 and we understand 1. Might we say that 2 is the number to represent the number of numbers we have thus far defined? Does that still count as a circular argument? Perhaps it would... If I went to count how many numbers we had, I would say "1... 2... So that's what 2 is!"

I guess this definition would be no better then me showing you two rocks and having you count them. This, by the way, is the kind of teaching that got us into this mess to begin with. It's really no different than you asking me "What is piety?" And I show you a series of things considered to be pious and saying "Piety is what all these things, stripped of their physical forms, still have in common."

Can we really say that is a fair way to define the pious? We would still know nothing of the pious, well, nothing we couldn't learn from a comprehensive list of pious things.

Of course, no one these days believes their is an objective piety, that it is only opinion. Might we one day have a similar belief about 2?

So, we're back to the problem "What is a 2?" Though, now I've acquired a new doubt that there even is such a thing as 2. How about this, what would I deny if I said there was no such thing as 2?

If you showed me 2 rocks and asked me to count them, shouldn't I have to say there are 2 rocks? I might try to weasle out of this by saying "There are 1+1" rocks." To which you would say "1+1 IS 2!"

So, in otherwords: . However, this is even still a sly way of using the concept of 2 to define the concept of 2. Allow me to explain...



I'm sleepy now and I'm about to get off. In case you didn't realize, I really just typed this out as I thought it up, please tell me what you think.

Hey Nervous.

A serious answer is 2 is anything you want it to be: it's just a label just like "cat" is a label to describe a particular kind of biological creature.

You can though characterize 2 as having some kind of ranking with respect to the other "labels" where if you look at whole numbers, -1 is "less than" 2 but 3 is "greater than two".

You can also define 2 to be part of some set where you can do algebra and arithmetic on with respect to other numbers.

Typically what we do in mathematics is that we have all these techniques and frameworks that operate on variables of some sort where the numbers themselves have no real interpretation, but in a specific problem, we give the numbers an interpretation.

A 2 by itself is meaningless, but if the 2 represents a physical constant, the height of a particular person, or something else with context, then this context helps give the 2 some meaning. However if you just said the number 2 to someone, they would say WTF and walk off.

This is not just how mathematics works, this is how all language works and also this is how computers work as well.

The analogy with computers is that you have 1's and 0's that mean absolutely nothing if you saw them (just like if you saw a 2). But if you give that data to a particular computer program, then the program will make its own "interpretation" of that data and use it for some specific use.

A computer game will treat it as an image or maybe text, audio, video, 3D models, animations or something else. A spreadsheet will treat it as cell data, formulas, charts and graphics. A zip file will treat it as a directory with compressed data.

All of these (and more) programs treat the data and interpret it in a different way, but the data is still just 1's and 0's in memory for all instances and the number "2" or whatever other number you have is also treated by us in whatever way we interpret it just as if we were the computer program and the numbers are the 1's and 0's in memory.

What about drawing a continuous line, and asserting that numbers represent a distance?

They don't have to represent a distance: this is just a particular kind of structure that is imposed on numbers.

The additional structures that add the idea of distance include metrics, and norms. If you want to add angle and geometry, you add an inner product. Inner products give you norms and metrics for free, norms give a metric for free but a metric is just a metric and doesn't give you anything else.

Not all things have special metric structure, but the numbers that many commonly use do like the one dimension sets including everything from the counting numbers (1 and above) all the way to the real numbers. The complex numbers have no real ordering in them like the reals, but you can impose inner products, norms, and metrics on these to get an idea of distance between two numbers as well as a relative relation between two numbers much in the same way that is done with vectors where you use inner product to get angle info and the norm to get distance info.

The most general form of representation is that of sets and sets don't need any kind of rank or comparison: you can add these if you want, but by no means is it necessary.

Chiro, you seem to be much more knowledgeable on the subject than I am. Though, it sounds like you are saying that 2 is like piety. It has no concrete meaning, it's just what we say of some things and not of other things. Am I understanding you right?

EDIT, after re-reading your post, it seems more like you are saying that 2 is simply 1<X<3 and that 1 is 0<X<2, and that 4 is 3<X<5, etc. Is this what you mean?

I don't know what that means. I've only learned up to differential calculus, and most of all that I've learned I had to teach myself. My school taught only algebra, never anything to do with sets. Or if it did, I don't remember.

Then I don't know what kind of answer you want. For one thing when you said "that 2 is simply 1<X<3 and that 1 is 0<X<2, and that 4 is 3<X<5, etc", that is certainly not true unless you add the information that x must be an integer- and then you would first have to define "integer".

Piano's definition of the non-negative integers defined a set of objects together with a 'successor function' satisfying certain axioms. "0" is defined as the single member of the set that is NOT the successor of any member (the axioms guarenteeing the existance of such a member), "1" is defined as the successor of "0", and "2" is defined as the successor of "1".

What johnsomeone gave is a specific instantiation of those axioms. "0" is defined to be the empty set and the successor function is defined by "the successor of x is the set containing x and all of its members. The "0"= {}, the empty set, "1", the successor of "0", is {{}}, the set whose only member is the empty set, and "2", the successor of "1", is {{}, {{}}}, the set containing 1= {{}} and {} which is contained in 1.

Originally Posted by Nervous

Chiro, you seem to be much more knowledgeable on the subject than I am. Though, it sounds like you are saying that 2 is like piety. It has no concrete meaning, it's just what we say of some things and not of other things. Am I understanding you right?

EDIT, after re-reading your post, it seems more like you are saying that 2 is simply 1<X<3 and that 1 is 0<X<2, and that 4 is 3<X<5, etc. Is this what you mean?

What I mean is that 2 is just a label in of itself: it has no more of an interpretation than "ASLKDJHASD" or 129837192837912 in an information theoretic way.

But what happens is that you give it an interpretation by the way you use it and the context within how its used.

When you treat it for example as real number or an integer, you are adding context. The context is added by the definition of real numbers, ordering, definitions of arithmetric and algebra and so on, but even with this you don't really know what 2 actually corresponds to non-mathematically.

So you add some more context and say that these values corresponds to heights in metres. You have added a little more context and now these things have units and you know they correspond to physical distances of some sort, but again you could go a bit further.

Then you say that they correspond to the height that a tennis ball travelled in 2 seconds when you dropped it off a building with a specific height on a specific day with specific environmental conditions and a bunch of other specifics.

This has added a tonne more context to the situation and now you are able to make relative interpretations based on what this number means in a variety of ways: you have gone from seeing a bunch of symbols to finally getting something that has a lot more meaning.

This is the nature of all information not just numbers. We take for granted that the structure of spoken and written language has a lot of implicit structure that gives all the symbols and the words context and the same is true to an extent for mathematics.

Happy to be corrected... my user name is true - it really has been a long time since I had to get my head around these problems (16 year old son with maths homework brings me back to it). However, enough rambling....

In answer to the question is 2 just a label, I'm pretty sure the answer is yes. sort of...

Nervous was pretty much on the right lines in his first post. I'll try to hack an explanation together... someone more eloquent may well have to tidy me up!

0 = the size of the set with no elements. the important thing here being that you are talking about the size of the set. Zero is the label that describes the null set
1, 2, 3, etc are descriptors of sets, each of which have a size of a number of elements that you can probably guess!

Then, when your head stops spinning, you accept that these labels are set sizes, and "2+2" gives you a set called "4" with... the number of elements in it you would expect from 4!! You did learn set theory at school - but you may not have been aware of it at the time :-)

And if you take this to its natural conclusion it also helps you understand Infinity. Infinity isn't a "number" you can count to - it's the size of a set. Erm - an infinite set! If you can get your head around this, then it helps make sense of statements like "one infinite set being larger than another"


Hope this helps... but having read my own post, I do fear for the quality of any of my son's homework I've tried to help him with!

Originally Posted by HallsofIvy

Piano's definition of the non-negative integers ...

Peano, Piano's key definition of the Naturals was too black and white to be useful.

.

the ancient greeks held that one was "unity" (a term that still finds currency in some areas of modern algebra). there is a certain "double-meaning" here: 1 represents not only "something" but also the (least) DIFFERENCE between "something" and "nothing". that is:

1 = 0+1: that is, 1 is "1 away from 0" (so 1 is not only an "object" but also a "distance"). here "+" doesn't mean our normal "addition" (although the notation IS suggestive), but rather just a way to indicate that we have "gone somewhere" (to 1) from "nowhere" (0). set-theorists like to use the notation s(0), where s stands for "successor".

for example, if you are counting sheep, 1 sheep is the smallest number (amount) of sheep you can have. but it is also the smallest difference two groups of (perhaps uncounted) sheep can have in number, and still not be "the same number".

in this view, 2 = 1+1 (here the "+" is STILL not addition, we just mean the NEXT "one" AFTER "one", so perhaps the set-theorists are right, and we should write:

2 = s(s(0)), which doesn't use some idea we haven't CREATED yet).

this marks an important first attempt at trying to define "what a number is" (or might be). numbers have an ORDER: first, second, third, fourth....(ignore, for the time being, the circularity in the adjectives, we could use something more abstract like:

|-th
||-th
|||-th
||||-th, and so on.

we can define numbers as:

"something you can count" (err..."one-by-one"), or:
"amounts of "units" you can compare. you can stack pennies, or pair the sheep from different flocks together, the larger flock is the one with "unmatched sheep".

the "counting numbers" (or NATURAL numbers, as they are more commonly called) are "made up of ones". we can do this with sets (this is just a fancy way of making "logical" what we do "intuitively")

0 = { } (an empty bag....nothing there, just the idea that something COULD be there, but it's not).
1 = {0} (the bag now has "something" in it, the empty bag)
2 = {{ },{0}} = {0,1} (we now have "different things to compare", so we have "duality"). basically all we're saying here is: 2 is 1 and 1. 2 is now "a set with two things in it".

we could write this 2 = {**}, but that leaves one wondering what the *'s are. but that's what we mean, essentially. take one orange, and another orange: that's two oranges.

add yet ANOTHER orange, you get:

3 = {0,1,2}, or 3 = {***}, or 3 = s(s(s(0)). it doesn't really matter what SYMBOL you use for 1, there's a certain similarity to:

|||

1&1&1

AAA

***

etc.

now "2" (the squiggle, for which we are indebted to the arabs, and their graceful script...although they got the idea from the hindus, who also have a flowing alphabet) is just an "abbreviation", because it gets cumbersome to express:

|||| & ||||| is |||||||||

(historical trivia: the ampersand "&" and the plus symbol "+" were originally the same symbol (which looked a bit like a freeway exchange) a stylized version of the Latin word "et", which means "and". the "=" is likewise "suggestive" and derived from drawing two parallel lines of the same length, an innovation championed by Robert Recorde who wrote (around 1557):

...to auoide the tediouſe repetition of theſe woordes : is equalle to : I will ſette as I doe often in woorke vſe, a paire of paralleles, or Gemowe lines of one lengthe, thus: =, bicauſe noe .2. thynges, can be moare equalle).

in simple terms: "two" is "one after one". as you can see this definition is NOT circular, because it only uses "one" and not "two". the use of the word "after" suggests a progression (in time and/or space), the idea that "growth" is an increase in physical dimension that occurs over time, is a very basic one in our language, and probably comes from "externalizing " our common internal experience of growing from a small infant to a larger adult. mathematical terms, as any other form of communication we have, serve the purpose of turning our internal experiences into external messages we can share. "2" is no different.

I'm not really a math person and have more questions than answers on this topic. But I think its an excellent topic in philosophy. Here is my understanding of the matter

1) Our ability to intuit 2 or more things comes from our ability to cognitively distinguish between things and recognize that things are different/separate. (at first your mother from the rest of reality, then gradually, step by step, you recognize intuitively other things)

2) This means that our understanding of separation (and therefore the root of counting) is directly related to our *perception* of reality. We can perceive things falsely.

3) With our ability speak, humans make and learn names for objects and concepts that share qualities. Every new word further expresses our desire to categorize what we are witnessing. Once things begin sharing definitions which we make either arbitrarily, intuitively, or in a calculated manner, we can count them together with other similar things (I think the definitive set of similarities is called cardinals or cardinality within in a mathematical set). For example, in counting how many tables I see, I'm not going to create them with positive synthesis adding up molecules that make them, but rather, would probably judge them by their purpose: a table has 4 legs and a top, it holds things up and is typically such and such dimensions so it fits in a room, and so I see 4 things that share these limits. When counting things such as tables, we can intuitively see them without making all the categorical calculations.

4) Animals share this ability to some degree. Watch an animal surrounded by predators move from side to side. He is intuiting and counting the dangers around him.

5) Words are not 100% definitive, they are categories in the philosophical realm and in mathematical terms are called sets. No words, categories, or sets have any meaning outside of context. So when we say 'water' we are not necessarily assuming that the consistency of water must be 100% H2o, else we wouldn't count it among the things we drink, but rather that water is a category and based on the context, it might be usable or not. Likewise when we ask for someone to pass the water, we're not asking them to levitate the water out of the carafe, rather, in speech we can make many generalizations. The same is true in mathematics. You cannot add things that don't share limits UNLESS YOU MEAN TO. You can count apples and oranges if your intention is to count the fruits on a table and that's what there is. If you intend to count only the apples among many fruits, then clearly that limits your set.

6) Ironically, in the physical world and possibly in the conceptual world, no two things can be 100% the same. Even two atoms side by side, assuming they could have the same subatomic structure, occupies different space in the past and have a different history of motion and interaction with reality - such as that they have collided with, and were repelled by different things in the vast universe. So again, at least as far as I understand, in the physical world, we never actually count two things that are 100% the same.

7) When dealing with theory, we have to make sure that there is sense behind the symbols and numbers we are putting together. On the one hand, there are limits to what we can know, and on the other, a limit is exactly what a set is - a thing with definable limits.

8) Even if the limits are beyond human grasp such as the set of {things that are infinite}, as beenalongtime said, we have created a category to count those things. Like in this example, you don't have to understand the specific definition of everything within the set of {a thing} in order to make a set. Many times realistically speaking, we don't know. For example, the set of {all things} includes the set of {all things not yet known} but not the specific cardinality of {all things not yet known} since its simply impossible for human comprehension.

I have more on the topic but would like to hear what you guys think on these definitions or if they are self evident.

Put a bean in a pot, that is "1." Put another bean in the pot, that is "2." etc, etc, etc.

Originally Posted by symbiosis

I'm not really a math person and have more questions than answers on this topic. But I think its an excellent topic in philosophy. Here is my understanding of the matter

1) Our ability to intuit 2 or more things comes from our ability to cognitively distinguish between things and recognize that things are different/separate. (at first your mother from the rest of reality, then gradually, step by step, you recognize intuitively other things)

2) This means that our understanding of separation (and therefore the root of counting) is directly related to our *perception* of reality. We can perceive things falsely.

3) With our ability speak, humans make and learn names for objects and concepts that share qualities. Every new word further expresses our desire to categorize what we are witnessing. Once things begin sharing definitions which we make either arbitrarily, intuitively, or in a calculated manner, we can count them together with other similar things (I think the definitive set of similarities is called cardinals or cardinality within in a mathematical set). For example, in counting how many tables I see, I'm not going to create them with positive synthesis adding up molecules that make them, but rather, would probably judge them by their purpose: a table has 4 legs and a top, it holds things up and is typically such and such dimensions so it fits in a room, and so I see 4 things that share these limits. When counting things such as tables, we can intuitively see them without making all the categorical calculations.

4) Animals share this ability to some degree. Watch an animal surrounded by predators move from side to side. He is intuiting and counting the dangers around him.

5) Words are not 100% definitive, they are categories in the philosophical realm and in mathematical terms are called sets. No words, categories, or sets have any meaning outside of context. So when we say 'water' we are not necessarily assuming that the consistency of water must be 100% H2o, else we wouldn't count it among the things we drink, but rather that water is a category and based on the context, it might be usable or not. Likewise when we ask for someone to pass the water, we're not asking them to levitate the water out of the carafe, rather, in speech we can make many generalizations. The same is true in mathematics. You cannot add things that don't share limits UNLESS YOU MEAN TO. You can count apples and oranges if your intention is to count the fruits on a table and that's what there is. If you intend to count only the apples among many fruits, then clearly that limits your set.

6) Ironically, in the physical world and possibly in the conceptual world, no two things can be 100% the same. Even two atoms side by side, assuming they could have the same subatomic structure, occupies different space in the past and have a different history of motion and interaction with reality - such as that they have collided with, and were repelled by different things in the vast universe. So again, at least as far as I understand, in the physical world, we never actually count two things that are 100% the same.

7) When dealing with theory, we have to make sure that there is sense behind the symbols and numbers we are putting together. On the one hand, there are limits to what we can know, and on the other, a limit is exactly what a set is - a thing with definable limits.

8) Even if the limits are beyond human grasp such as the set of {things that are infinite}, as beenalongtime said, we have created a category to count those things. Like in this example, you don't have to understand the specific definition of everything within the set of {a thing} in order to make a set. Many times realistically speaking, we don't know. For example, the set of {all things} includes the set of {all things not yet known} but not the specific cardinality of {all things not yet known} since its simply impossible for human comprehension.

I have more on the topic but would like to hear what you guys think on these definitions or if they are self evident.

I agree that the idea of "two" arises from our basic perceptual distinction between "this" and "that". this cognitive ability, the recognition of similarity and difference, seems to be basic to most higher central nervous system functions (and perhaps even realized via chemical markers on the cellular level). We have taken this perceptual ability, and run with it: we have a whole host of linguistic terms (and mathematical terms, as well) for: "same, but not the same" (for example: similar, equivalent, like, etc.). in other words, we seem to have an innate ability to ABSTRACT: two apples may not be identical, but we can treat them in our THOUGHT as if they were (hmm...it turns out i am contemplating a purely hypothetical apple even as i type this).

when one is first learning language (an undeniable boon to us expressing our internal capacity for thought), one often learns by EXAMPLE: "this is an apple", "this is a chair", etc. "how many" is a refinement of the idea of "how much", which rests on us being able to distinguish different "amounts".

perhaps the REASON we devised set theory, is that it mimics, in some fashion, the internal decision procedure in our brains. our brains appear to have an innate ability to SYMBOLIZE (something we experience in a quite interesting fashion when we dream). the word "two" (or any of its counter-parts "deux", "dos", "zwei", or the proto-indo-european "dvi") appears to be one of our oldest words. in other words, no matter what we declare "2" to mean MATHEMATICALLY, it appears we have some a priori idea of what we WANT to define before we even start (like "two" is what signifies how may eyes, or ears, or hands we have).

it's...hard...to talk about these sorts of things "truthfully", because even words aren't the things they represent. i cannot convey accurately the nature of how i UNDERSTAND what "two" is, i have to trust that YOU have the same understanding as you read this: a very large leap of faith, indeed. it seems to work: when i order 2 coffees from the coffeeshop, that is indeed what i receive (usually placing my order by holding up 2 fingers, an ingenious short-circuiting of linguistic error).

that is: on a HUMAN level (not a mathematical, rigorously logical one), when i say "2", i expect you to figure out on your own what two upraised fingers and:

||

have in common. and most everyone does. is it magic?