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[SirPeebles]

As for proving that 1 is positive.

We know that either 1 is positive, 1 is zero, or -1 is positive.  Let's suppose that 1 is not positive, and look for a contradiction.  Since 1 is not positive, either 1 is zero or -1 is positive.  But 1 cannot be zero by hypothesis.  So all we need to do is rule out the possibility that -1 is positive.

Suppose that -1 were positive.  Well, positive x positive = positive, so then -1 x -1 is a positive number.  But I claim that -1 x -1 = 1.  Once we see why, then we will have our contradiction, since we are currently supposing that 1 is not positive.  To check that a number is equal to 1, it suffices to check that its sum with -1 is 0.

(-1 x -1) + -1 = (-1 x -1) + (-1 x 1) = -1 x (-1 + 1) = -1 x 0

Above we first used that 1 is the multiplicative identity, then the distributive law, and then the fact that -1 is the additive inverse of 1.

What is left is to show that -1 x 0 = 0.  In fact, a x 0 = 0 for any number a, as I'll now show:

a = a x 1 = a x (1 + 0) = (a x 1) + (a x 0) = a + (a x 0).

Now add -a to both sides:

-a + a = -a + ( a + (a x 0))
0 =  (-a + a) + (a x 0)
0 = 0 + (a x 0)
0 = a x 0.

Thus above we had that (-1 x -1) + -1 = 0, so -1 x -1 = 1.  This gives us our contradiction.

[Kirian]

As for proving that 1 is positive.

We know that either 1 is positive, 1 is zero, or -1 is positive.  Let's suppose that 1 is not positive, and look for a contradiction.  Since 1 is not positive, either 1 is zero or -1 is positive.  But 1 cannot be zero by hypothesis.  So all we need to do is rule out the possibility that -1 is positive.

Suppose that -1 were positive.  Well, positive x positive = positive, so then -1 x -1 is a positive number.  But I claim that -1 x -1 = 1.  Once we see why, then we will have our contradiction, since we are currently supposing that 1 is not positive.  To check that a number is equal to 1, it suffices to check that its sum with -1 is 0.

(-1 x -1) + -1 = (-1 x -1) + (-1 x 1) = -1 x (-1 + 1) = -1 x 0

Above we first used that 1 is the multiplicative identity, then the distributive law, and then the fact that -1 is the additive inverse of 1.

What is left is to show that -1 x 0 = 0.  In fact, a x 0 = 0 for any number a, as I'll now show:

a = a x 1 = a x (1 + 0) = (a x 1) + (a x 0) = a + (a x 0).

Now add -a to both sides:

-a + a = -a + ( a + (a x 0))
0 =  (-a + a) + (a x 0)
0 = 0 + (a x 0)
0 = a x 0.

Thus above we had that (-1 x -1) + -1 = 0, so -1 x -1 = 1.  This gives us our contradiction.

This looks suspiciously like the specious proof that 1 = 2 (you know, the one that divides by zero a couple times and obfuscates it).

[Watno]

Well, the successor function is often considered to be more basic than addition, when it comes to the natural numbers (0, 1, 2, 3, ...).

A common axiomization of these numbers states, among things, that every number has a unique successor, and that every number is the successor of some unique other number, with a single exception.  That single number which is not the successor of another we name 0.  1 is the defined as the successor of 0.  2 as the successor of 1.  Addition is defined in terms of this simpler operation.

Isn't addition defined like this in that model?

0 + n := n
Succ (m) + n := Succ(m + n)

So in that case it is obvious that 1+1= Succ(0)+Succ(0)=Succ(0+Succ(0))=Succ(Succ(0))=2 ?

[SirPeebles]

Well, the successor function is often considered to be more basic than addition, when it comes to the natural numbers (0, 1, 2, 3, ...).

A common axiomization of these numbers states, among things, that every number has a unique successor, and that every number is the successor of some unique other number, with a single exception.  That single number which is not the successor of another we name 0.  1 is the defined as the successor of 0.  2 as the successor of 1.  Addition is defined in terms of this simpler operation.

Isn't addition defined like this in that model?

0 + n := n
Succ (m) + n := Succ(m + n)

So in that case it is obvious that 1+1= Succ(0)+Succ(0)=Succ(0+Succ(0))=Succ(Succ(0))=2 ?

Probably something like that.  Then it is a bit of work to show that addition is associative and commutative.  In other words, that (x + y) + z = x + (y + z)  and that x + y = y + x.

[Watno]

I don't think I used associativity and commutativity?

[SirPeebles]

Sure, I didn't mean to imply that you did.  I was just drawing attention to the fact that commutativity and associativity are non obvious from this formal definition.  As opposed to the intuitive idea of 2+3 means put two apples into a bag, then three apples, and count how many you have.

[WanderingWinder]

There is no way to make the Complex Numbers into an Ordered Field (with the standard field operators), which is why there's no canonical total order. However, if you don't care about relating the ordering to the field algebra, then you're free to use whichever order you choose. I'm partial lexicographic orderings, myself. I mean, if you're not going to relate it to the field, might as go all the way.

My first homework problem in my first graduate math course was to prove that 1 is positive.

Wait, couldn't you just say that 1 > 0, so it is positive?

Most people could; someone in a graduate math course can't.

From this, I logically deduce that entering a graduate math course makes you less intelligent



Seriously, this kind of thing is what I hate about some of this formal math - you come up with some obscure, strange 'proof' that 1>0, but what it rests on is not at all more intuitive than the conclusion 1>0 which it is supposedly providing a basis for.

[Watno]

But what is it based on in significantly less.

[AJD]

Seriously, this kind of thing is what I hate about some of this formal math - you come up with some obscure, strange 'proof' that 1>0, but what it rests on is not at all more intuitive than the conclusion 1>0 which it is supposedly providing a basis for.

Really what's being proven in these case is these axioms are sufficient to describe a system in which 1 > 0; we don't need to stipulate that 1 > 0 a priori in order to state the fundamental properties of the number system (and therefore there isn't some other possible system which shares all those other properties but not the property that 1 > 0).

[clb]

What's the definition of 2?
I thought it was defined as 1+1.

Indeed it is. Most of the counter examples badly define either counting or the arithmetic operator. For example, 1 river + 1 river = 1 bigger river.

This reminds me of my friend's father. He asserts that a sandwich cut into two (probably triangles) is not two half-sandwiches, but rather two entire (albeit admittedly smaller) sandwiches. However, a single piece of bread folded over [stuff put on sandwiches] is a half sandwich. It is frustrating that it almost makes sense, but then doesn't.

[SirPeebles]

There is no way to make the Complex Numbers into an Ordered Field (with the standard field operators), which is why there's no canonical total order. However, if you don't care about relating the ordering to the field algebra, then you're free to use whichever order you choose. I'm partial lexicographic orderings, myself. I mean, if you're not going to relate it to the field, might as go all the way.

My first homework problem in my first graduate math course was to prove that 1 is positive.

Wait, couldn't you just say that 1 > 0, so it is positive?

Most people could; someone in a graduate math course can't.

From this, I logically deduce that entering a graduate math course makes you less intelligent



Seriously, this kind of thing is what I hate about some of this formal math - you come up with some obscure, strange 'proof' that 1>0, but what it rests on is not at all more intuitive than the conclusion 1>0 which it is supposedly providing a basis for.

It is true that we already have a strong intuition.

Two points:

1)  This is really about showing that in every ordered field, the number 1 is a positive element.  You can think of this either as proving an interesting fact about ordered fields, or as evidence that the concept of an ordered field really is matching our intuition.  There are ordered fields which come up which are not just subsets of the real numbers.

2)  Perhaps even more important historically, ordered fields show up in the axiomization of the real numbers.  Somehow, we humans have quite a bit of intuition for some rather bizarre numbers.  Not just whole numbers, but fractions, the square root of two, pi, the cosine of a 72 degree angle, and so forth.  We can list lots of "obvious" facts about real numbers:  any number times zero is zero, the product of two negative numbers is positive, every positive number has a square root, every cubic equation has at least one solution, there is no largest number, between any two distinct numbers there is a third number, ... .  Lots and lots of facts.  If I want to prove something about real numbers, how many of these intuitive facts do I need to know?  In physics, if you can't solve a problem, sometimes you need to go observe the physical world to discover new truths.  Is it the same for the study of real numbers?  Well, it turns out that our intuition informs us that the real numbers form a complete ordered field, and then we can prove mathematically that there is only one complete ordered field.  Thus, the basic axioms of a complete ordered field must be all that one needs to begin a comprehensive study of the real numbers.  In particular, one is able to recover intuitive facts such as those I listed above.

[WanderingWinder]

But what is it based on in significantly less.

I think what you're trying to say is "But what it is based on is significantly less", which I can say, uh, no, no it isn't. Or I guess, define "less", and I'm sure you're doing it in a way which is different from mine.

Seriously, this kind of thing is what I hate about some of this formal math - you come up with some obscure, strange 'proof' that 1>0, but what it rests on is not at all more intuitive than the conclusion 1>0 which it is supposedly providing a basis for.

Really what's being proven in these case is these axioms are sufficient to describe a system in which 1 > 0; we don't need to stipulate that 1 > 0 a priori in order to state the fundamental properties of the number system (and therefore there isn't some other possible system which shares all those other properties but not the property that 1 > 0).

Okay... but the problem is, (well first of all, who cares, we don't live in some fairy world of different mathematics), but more importantly, if there WERE some contradiction, it would show the axioms wrong rather than showing that 1>0 is possibly false in some universe or number system or whatever. i.e. you're proving things backwards.

Okay, so you can prove that these things logically prove the other thing. Why is this considered an achievement? It would be like asking to prove, given that I have typed this, that I exist.

[WanderingWinder]

1)  This is really about showing that in every ordered field, the number 1 is a positive element.  You can think of this either as proving an interesting fact about ordered fields, or as evidence that the concept of an ordered field really is matching our intuition.  There are ordered fields which come up which are not just subsets of the real numbers.

Can I think of it as someone coming up with a definition to match common understanding then patting themself on the back for having that match common intuition?



Seriously, the logical argument that "Ordered fields match our intuition about real numbers" -> "Ordered fields have applications beyond the real numbers" -> "Ordered fields can thus provide us with an understanding which surpasses our intuition" is really bad. It just IS your intuition and/or you actually don't have any evidence at all, because you haven't established its applicability beyond the realm which you originally verified it on.

[sudgy]

I think one of the points of math is to start with as few axioms as possible, and prove anything else from that.  If we can prove 1 > 0, then we don't have to postulate it.

I remember reading a proof (that was really simple) that with any line segment, you can create an equilateral triangle with sides that long.  Now, the conclusion is obvious, but if we can prove it with other axioms, we don't need to make that an axiom.

[WanderingWinder]

I think one of the points of math is to start with as few axioms as possible, and prove anything else from that.  If we can prove 1 > 0, then we don't have to postulate it.

Why? Shouldn't the point be to not have as numerically few axioms as possible (which also brings up the disturbing question of what counts as separate axioms), but rather to have things rest on as sure a set of 'givens' as possible? I mean, this isn't a Math question at all, it's an epistemology question.

I remember reading a proof (that was really simple) that with any line segment, you can create an equilateral triangle with sides that long.  Now, the conclusion is obvious, but if we can prove it with other axioms, we don't need to make that an axiom.

But this doesn't actually gain us anything. Why did we have to prove it in the first place? What is wrong with making something obviously true an axiom? If there is something wrong with making something obviously true an axiom, how does that not invalidate the whole process of making axioms?
Didn't you take it as an additional axiom (and one which actually seems most probably wrong to me) that it is better to have fewer axioms?

[sudgy]

Well, as soon as we know that a theorem can be proven with our chosen set of axioms, we can act like the theorem is an axiom.  The reason to have a few axioms is that we don't need to list some insane number of axioms.  Axioms are supposed to be the bare minimum of points required to show everything.

Also, we need to have some things that are proved.  Certain theorems, though true, don't make much intuitive sense (and some to certain people more than others).  Where do you draw the line of what needs to be proven and what not?

[SirPeebles]

It really is quite an accomplishment that the theory of real numbers can be reduced to a finite number of fundamental properties.

Perhaps to gain an appreciation for this utterly nonobvious fact, it would be nice to look at an example where it doesn't happen.  Perhaps the most famous example was in geometry, regarding Euclid's postulates.  Eucild attempt to reduce all of geometry to just a short list of axioms.  The plan was to have a short list of basic assumed facts that essentially all humans would agree are true.  Now yes, what is obvious to one person may not be obvious to another.  But that's the point really.  Euclid began from observations like "for any two points, there is a straight line which joins them"  rather than something about inscribed angles.  But ultimately he included one postulate which seemed not quite as evident, the parallel postulate.  For centuries mathematicians tried to reduce the parallel postulate to more intuitive statements, but failed.  Eventually the 18th and 19th centuries, people discovered that there were geometries -- models of the original four postulates -- where the parallel postulate did not hold.  This led to the discoveries of spherical, hyperbolic, and ultimately Riemannian geometry.  20th century advances in physics has revealed that in the actual universe, the parallel postulate isn't obeyed -- space is locally curved.

Another instance is in set theory.  Mathematicians began studying set theory in earnest in the 19th century.  Many fundamental axioms have been agreed upon:  the empty set exists, the union of two sets exists, ...  But then there are statements like the Continuum Hypothesis.  This statement is too complicated for most people to believe that it is obviously true or obviously false.  So you could try to prove or disprove it from the agreed upon axioms.  The trouble is, mathematicians have been able to show that it is independent from the communally agreed upon axioms.  That is, they constructed one model of set theory where the Continuum Hypothesis is obeyed, and a different model of set theory where the Continuum Hypothesis is violated.

This does not happen in the theory of real numbers.  Any two complete ordered fields are isomorphic.  It's not as though there is going to be one complete ordered field where every 17th degree polynomial has a root and another complete ordered field where there is a 17th degree polynomial without a root, thereby requiring us to go back to our evolved reason and intuition to divine which one of these two properties is valid for the true real numbers.  I can build a fancy-pants complete ordered field where every 17th degree polynomial has a root.  The true real numbers is the only complete ordered field.  So I have proved something for the true real numbers.  You can't argue that your personal conception of the real numbers might have a 17th degree polynomial with no roots unless you are willing to reject that the real numbers form a complete ordered field -- that is, unless you are willing to reject one of a short list of "obvious" facts.

[TWoos]

There is no way to make the Complex Numbers into an Ordered Field (with the standard field operators), which is why there's no canonical total order. However, if you don't care about relating the ordering to the field algebra, then you're free to use whichever order you choose. I'm partial lexicographic orderings, myself. I mean, if you're not going to relate it to the field, might as go all the way.

My first homework problem in my first graduate math course was to prove that 1 is positive.

Wait, couldn't you just say that 1 > 0, so it is positive?

Most people could; someone in a graduate math course can't.

In certain graduate level math classes you can't.  It's perfectly valid in others.  It all depends on how fundamental the class gets.

[TWoos]

I think one of the points of math is to start with as few axioms as possible, and prove anything else from that.  If we can prove 1 > 0, then we don't have to postulate it.

Why? Shouldn't the point be to not have as numerically few axioms as possible (which also brings up the disturbing question of what counts as separate axioms),

What counts as a separate axiom is what you define as a separate axiom. There's nothing disturbing about it.  Axioms are specified.

I remember reading a proof (that was really simple) that with any line segment, you can create an equilateral triangle with sides that long.  Now, the conclusion is obvious, but if we can prove it with other axioms, we don't need to make that an axiom.

But this doesn't actually gain us anything. Why did we have to prove it in the first place? What is wrong with making something obviously true an axiom?

Because when you create a new system of numbers (and yes, you can do this, and yes, there are reasons you might want to), you need to prove that every axiom that applied to the old system applies to the new system in order to use the proofs derived from those axioms.  The fewer axioms you used, the less you have to prove.

If there is something wrong with making something obviously true an axiom, how does that not invalidate the whole process of making axioms?
Didn't you take it as an additional axiom (and one which actually seems most probably wrong to me) that it is better to have fewer axioms?

No, it is not an axiom that it is better to have fewer axioms.  That's planning for future laziness, and mathematicians are all about being lazy.

[PSGarak]

Intuition has limits--it's of no use when we're faced with situations dissimilar to what we have faced before. Logic has different limits--it only works on things that are well-defined, because otherwise you get caught in meaningless semantic arguments. The strength of one approach covers for the weakness of the other, if we can direct both at a single problem.

Axiomatizing systems is an approach for taking what we know from intuition, and using logic to extend that knowledge to cover things we don't have experience for yet. There are other approaches, but they're fuzzier.

Axiomatic systems have two major weaknesses: They're fragile, and they can be unwieldy. By fragile, I mean that classical logic doesn't deal with consistency well. Like, at all. Like, a single inconsistency, no matter how small, renders the entire system invalid. By unwieldy, I mean that systems with large numbers of axioms are difficult to reason about, because they have more stuff in them.

Both of these problems can be alleviated by reducing the number of axioms you're dealing with. The more that you can reduce "intuitive axioms" to "theorems" by proving them from other axioms, the simpler your system becomes. This makes it less likely to be inconsistent, and easier to work with because there's less facts to keep straight.

Simpler systems also make it easier to test things that you don't know if they're true or not. You can try to test if a fact is true by seeing if it interacts with other axioms in a way that creates a contradiction. In simpler systems, it's much easier to pinpoint which actual a contradiction arose from, and then you can evaluate which of the two axioms you want to discard.

Even in an example as synthetic as the real number line, this gets important when e.g. you want to start talking about infinities. Infinity is inherently a non-intuitive concept, so any attempt to intuit how it ought to behave is going to lead to contradictions. If your foundation is a giant pile of "intuitive facts" about how numbers behave, it's really hard to add infinity to the mix and keep them all straight. You'd have to gut and replace like half your damn system in order to work in infinity, and even then you won't be sure you've done a complete job. When your system is built on like 5 rules, it's easy to change one rule to admit infinity, and then study the consequences in a logical fashion.

[SirPeebles]

i = sqrt(-1) = sqrt(1/-1) = 1/sqrt(-1) = 1/i

Since we've shown i = 1/i, we conclude that i^2 = 1

Thus 1 = -1. 

Whoops, guess 1 is negative after all.

[sudgy]

i = sqrt(-1) = sqrt(1/-1) = 1/sqrt(-1) = 1/i

Since we've shown i = 1/i, we conclude that i^2 = 1

Thus 1 = -1. 

Whoops, guess 1 is negative after all.

Argh, where is the contradiction?

[AJD]

i = sqrt(-1) = sqrt(1/-1) = 1/sqrt(-1) = 1/i

Since we've shown i = 1/i, we conclude that i^2 = 1

Thus 1 = -1. 

Whoops, guess 1 is negative after all.

Argh, where is the contradiction?

I think the contradiction is that the operator sqrt() isn't well-defined; sqrt(–1) can evaluate to either i or –i. So

i = 1/sqrt(–1)

is true, but only if the sqrt(–1) on the right side there is –i, not i.

[AJD]

Seriously, this kind of thing is what I hate about some of this formal math - you come up with some obscure, strange 'proof' that 1>0, but what it rests on is not at all more intuitive than the conclusion 1>0 which it is supposedly providing a basis for.

Really what's being proven in these case is these axioms are sufficient to describe a system in which 1 > 0; we don't need to stipulate that 1 > 0 a priori in order to state the fundamental properties of the number system (and therefore there isn't some other possible system which shares all those other properties but not the property that 1 > 0).

Okay... but the problem is, (well first of all, who cares, we don't live in some fairy world of different mathematics), but more importantly, if there WERE some contradiction, it would show the axioms wrong rather than showing that 1>0 is possibly false in some universe or number system or whatever. i.e. you're proving things backwards.

Well, again, the real numbers aren't the only thing mathematicians are interested in. What this shows is that any system that resembles the real numbers in conforming to the axioms that define an ordered field also resembles it in having 1 > 0.

[WalrusMcFishSr]