## Really basic stuff that was never proven in class

For the discussion of math. Duh.

Moderators: gmalivuk, Moderators General, Prelates

mmmcannibalism
Posts: 2150
Joined: Tue Jun 30, 2009 6:16 am UTC

### Really basic stuff that was never proven in class

Just a thread to discuss basic proofs for things that are often taught at some level in school without a proper proof. For instance, foil multiplication of polynomials is taught in algebra 1, but at least in my school it was never properly explained why(beyond you multiply each term in the first by each term of the last).
Attachments
Izawwlgood wrote:I for one would happily live on an island as a fuzzy seal-human.

Oregonaut wrote:Damn fetuses and their terroist plots.

lemma
Posts: 138
Joined: Sat May 10, 2008 2:41 pm UTC

### Re: Really basic stuff that was never proven

Oh, that's awesome If only I taught the younger kids so that I could use it with them!

Dason
Posts: 1311
Joined: Wed Dec 02, 2009 7:06 am UTC
Location: ~/

### Re: Really basic stuff that was never proven

Our teacher just showed us it as a basic result of the distributive property:

Code: Select all

(a+b)*(c+d)                  | (a+b) is just a number so we can distribute(a+b)*c + (a+b)*d            | Now we can use the basic distributive propertya*c+b*c + a*d+b*d

rearrange if you feel the need. I think the picture your provide gives a good geometric interpretation of why it holds though.
double epsilon = -.0000001;

mike-l
Posts: 2758
Joined: Tue Sep 04, 2007 2:16 am UTC

### Re: Really basic stuff that was never proven

Dason wrote:Our teacher just showed us it as a basic result of the distributive property:

Code: Select all

(a+b)*(c+d)                  | (a+b) is just a number so we can distribute(a+b)*c + (a+b)*d            | Now we can use the basic distributive propertya*c+b*c + a*d+b*d

rearrange if you feel the need. I think the picture your provide gives a good geometric interpretation of why it holds though.

Yes, but why does the distributive property hold? (Same picture, just with only one row)
addams wrote:This forum has some very well educated people typing away in loops with Sourmilk. He is a lucky Sourmilk.

Kurushimi
Posts: 841
Joined: Thu Oct 02, 2008 12:06 am UTC

### Re: Really basic stuff that was never proven

In order to answer that you have to define multiplication.

And the whole "repeated addition" thing isn't going to cut it. What about fractions? What are we doing when we multiply fractions?

mike-l
Posts: 2758
Joined: Tue Sep 04, 2007 2:16 am UTC

### Re: Really basic stuff that was never proven

Kurushimi wrote:In order to answer that you have to define multiplication.

And the whole "repeated addition" thing isn't going to cut it. What about fractions? What are we doing when we multiply fractions?

How do you define fractions without multiplication? You generally start with natural numbers, define multiplication as repeated addition, then define negatives and fractions as pairs with equivalence relations.
addams wrote:This forum has some very well educated people typing away in loops with Sourmilk. He is a lucky Sourmilk.

jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5967
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

### Re: Really basic stuff that was never proven

mike-l wrote:Yes, but why does the distributive property hold? (Same picture, just with only one row)

I think we proved the rationals were a field at some point, but I don't think we ever made the final step to the reals (that step being "arithmetic operations are continuous"). Not teaching field axioms (and matrix arithmetic) is a real deficit of most advanced high school maths educations. You can also use them to prove that (-a)*(-b) = a*b for instance.

There's also a bit of a problem with the diagram if one of A and B is negative, but not the other, though you could rectify it with a case bash.

Probably most people see a proof of pythagoras and a proof of the volume formula for a sphere just not the best proofs of them.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

Cleverbeans
Posts: 1378
Joined: Wed Mar 26, 2008 1:16 pm UTC

### Re: Really basic stuff that was never proven

Pretty sure this is covered in Book I of Euclid's elements.
"Labor is prior to, and independent of, capital. Capital is only the fruit of labor, and could never have existed if labor had not first existed. Labor is the superior of capital, and deserves much the higher consideration." - Abraham Lincoln

Aiwendil42
Posts: 133
Joined: Mon May 17, 2010 8:52 pm UTC

### Re: Really basic stuff that was never proven

Pretty sure this is covered in Book I of Euclid's elements.

You've misunderstood the purpose of this thread. It's for basic things that posters were taught, but not given proofs for - not for basic things that have never been proven at all.

mmmcannibalism
Posts: 2150
Joined: Tue Jun 30, 2009 6:16 am UTC

### Re: Really basic stuff that was never proven

There's also a bit of a problem with the diagram if one of A and B is negative, but not the other, though you could rectify it with a case bash.

It's still easy to demonstrate with geometry

edit--yeah I messed up that picture when I thought about it, the top part is (b-a)(c+d)
Attachments
Last edited by mmmcannibalism on Fri Aug 20, 2010 11:50 pm UTC, edited 1 time in total.
Izawwlgood wrote:I for one would happily live on an island as a fuzzy seal-human.

Oregonaut wrote:Damn fetuses and their terroist plots.

squareroot
Posts: 548
Joined: Tue Jan 12, 2010 1:04 am UTC
Contact:

### Re: Really basic stuff that was never proven

Fundamental theorem of arithmetic. I'd asked why it was true when I was six, I never got a satisfactory response. I got things like "Well, can you think of any examples?" and crap I like that. I had to wait until I got a book of number theory before I actually saw a proof.
<signature content="" style="tag:html;" overused meta />
Good fucking job Will Yu, you found me - __ -

Seraph
Posts: 343
Joined: Mon Jul 16, 2007 4:51 pm UTC

### Re: Really basic stuff that was never proven

mmmcannibalism wrote:
There's also a bit of a problem with the diagram if one of A and B is negative, but not the other, though you could rectify it with a case bash.

It's still easy to demonstrate with geometry

If it's easy then why did you screw it up?

Let B = 2, C = 3, D = 4, A = -1.
(A + B)*(C + D) = 1 * 7 = 7

(A + B)*(C + D) = BC + BD = 2*3 + 2*4 = 6 + 8 = 14

14 != 7

Dason
Posts: 1311
Joined: Wed Dec 02, 2009 7:06 am UTC
Location: ~/

### Re: Really basic stuff that was never proven

Seraph wrote:
mmmcannibalism wrote:
There's also a bit of a problem with the diagram if one of A and B is negative, but not the other, though you could rectify it with a case bash.

It's still easy to demonstrate with geometry

If it's easy then why did you screw it up?

Let B = 2, C = 3, D = 4, A = -1.
(A + B)*(C + D) = 1 * 7 = 7

(A + B)*(C + D) = BC + BD = 2*3 + 2*4 = 6 + 8 = 14

14 != 7

To be fair the formula he has at the bottom just says area. The area of the whole thing would be ((b-a)+a)*(c+d) which does equal bc+bd. If you want to do (b-a)*(c+d) then you just sum the top two squares to get the desired result.
double epsilon = -.0000001;

cristobal
Posts: 7
Joined: Sun Jan 03, 2010 8:26 am UTC

### Re: Really basic stuff that was never proven

To be really pedantic, no one ever proved to me that the total area of a shape is equal to the sum of the areas of the partition. This is a non-trivial result.

dean.menezes
Posts: 135
Joined: Sat Nov 15, 2008 3:47 am UTC

### Re: Really basic stuff that was never proven

squareroot wrote:Fundamental theorem of arithmetic. I'd asked why it was true when I was six, I never got a satisfactory response. I got things like "Well, can you think of any examples?" and crap I like that. I had to wait until I got a book of number theory before I actually saw a proof.

They were probably trying to get you to learn something.That theorem is absolutely trivial.
Attachments
foo1.png (8.96 KiB) Viewed 6652 times

TiglathPileser3
Posts: 11
Joined: Tue Jun 22, 2010 3:18 pm UTC

### Re: Really basic stuff that was never proven

Whenever people define pi to me they say it is "the ratio of a circle's circumference to its diameter." I always thought that it wasn't intuitively obvious that this is a constant.

dean.menezes
Posts: 135
Joined: Sat Nov 15, 2008 3:47 am UTC

### Re: Really basic stuff that was never proven

Divide the circle into [imath]n[/imath] sectors and make them into triangles. Then take the limit as the number of triangles approaches infinity.
Attachments
lol.GIF (3.83 KiB) Viewed 6632 times

letterX
Posts: 535
Joined: Fri Feb 22, 2008 4:00 am UTC
Location: Ithaca, NY

### Re: Really basic stuff that was never proven

TiglathPileser3 wrote:Whenever people define pi to me they say it is "the ratio of a circle's circumference to its diameter." I always thought that it wasn't intuitively obvious that this is a constant.

It's not. In particular, there is no such thing as 'pi' in non-euclidean geometries. The ratio of the circumference to the diameter will depend on how big your 'circle' is.

Kurushimi
Posts: 841
Joined: Thu Oct 02, 2008 12:06 am UTC

### Re: Really basic stuff that was never proven

TiglathPileser3 wrote:Whenever people define pi to me they say it is "the ratio of a circle's circumference to its diameter." I always thought that it wasn't intuitively obvious that this is a constant.

Well, it's intuitive if you remember that all circles are similar to each other and all parts of a similar shape are grown/shrank in the same proportion.

squareroot
Posts: 548
Joined: Tue Jan 12, 2010 1:04 am UTC
Contact:

### Re: Really basic stuff that was never proven

They were probably trying to get you to learn something.That theorem is absolutely trivial.

Yeah, you go over to your nearest elementary school and try to prove that to them. And I think the fact that gcd(n,m)=1 -> (there exists r and s such that rn+sm=1) is much less elegant to show - if you can call the proof you gave elegant. Of course it seems trivial now, I just had trouble with the fact that (p does not divide n) && (p divides nm) -> (p divides m). It sound so simple it's silly, but that's just because we naturally think of integers' unique factorization. I actually got a bit of relief when I learned that unique factorization fails on some fields, and that I wasn't just being paranoid/stupid for those seven years of my life.
<signature content="" style="tag:html;" overused meta />
Good fucking job Will Yu, you found me - __ -

dean.menezes
Posts: 135
Joined: Sat Nov 15, 2008 3:47 am UTC

### Re: Really basic stuff that was never proven

squareroot wrote:Yeah, you go over to your nearest elementary school and try to prove that to them. And I think the fact that gcd(n,m)=1 -> (there exists r and s such that rn+sm=1) is much less elegant to show - if you can call the proof you gave elegant. Of course it seems trivial now, I just had trouble with the fact that (p does not divide n) && (p divides nm) -> (p divides m).

That was mostly a parody of how mathematicians call proven theorems no matter how difficult it was to prove it in the first place, which is why I put the smiley.

Tirian
Posts: 1891
Joined: Fri Feb 15, 2008 6:03 pm UTC

### Re: Really basic stuff that was never proven

About a year ago, for lulz I undertook proving that multiplication of the natural numbers is commutative. Holy cow. That was a half-page of dense symbolic manipulation even with the assumption that multiplication is associative and addition is associative and commutative. I can appreciate why people take it for granted.

WarDaft
Posts: 1583
Joined: Thu Jul 30, 2009 3:16 pm UTC

### Re: Really basic stuff that was never proven

Very rarely (I would venture perhaps even never) has the proof been given that 1+1 does in fact equal 2 before more advanced topics are covered.
All Shadow priest spells that deal Fire damage now appear green.
Big freaky cereal boxes of death.

squareroot
Posts: 548
Joined: Tue Jan 12, 2010 1:04 am UTC
Contact:

### Re: Really basic stuff that was never proven

dean.menezes wrote:
squareroot wrote:Yeah, you go over to your nearest elementary school and try to prove that to them. And I think the fact that gcd(n,m)=1 -> (there exists r and s such that rn+sm=1) is much less elegant to show - if you can call the proof you gave elegant. Of course it seems trivial now, I just had trouble with the fact that (p does not divide n) && (p divides nm) -> (p divides m).

That was mostly a parody of how mathematicians call proven theorems no matter how difficult it was to prove it in the first place, which is why I put the smiley.

Oh, I'm sorry. I was just feeling some old steam toward the people who never actually explained it to me. I didn't catch your humor.

For the record, addition's associativity and commutativity are trivial - inductive reasoning gives commutativity away almost for free, and associativity is also pretty simple.

To prove commutativity... Well, I think I would prove distributive property first.

a*(b+c) = (a-1)*(b+c) + b + c = (a-2)*(b+c) + b + c + b +c = (a-2)*(b+c) + 2*b + 2*c.... then use inductive reasoning and collect terms of b and c, with base cases (0)*(b+c) + a*b + a*c = a*b + a*c and (1)*(b+c) + (a-1)*b + (a-1)*c = a*b + a*c. Once you have that, proving commutativity is a lot easier.
<signature content="" style="tag:html;" overused meta />
Good fucking job Will Yu, you found me - __ -

silverhammermba
Posts: 178
Joined: Fri Oct 13, 2006 1:16 am UTC

### Re: Really basic stuff that was never proven

dean.menezes wrote:
squareroot wrote:Fundamental theorem of arithmetic. I'd asked why it was true when I was six, I never got a satisfactory response. I got things like "Well, can you think of any examples?" and crap I like that. I had to wait until I got a book of number theory before I actually saw a proof.

They were probably trying to get you to learn something.That theorem is absolutely trivial.

Please tell me that the proof you posted is intentionally obfuscated. There are truly trivial proofs of the fundamental theorem of arithmetic that don't require all of that complexity.

antonfire
Posts: 1772
Joined: Thu Apr 05, 2007 7:31 pm UTC

### Re: Really basic stuff that was never proven

Give one.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

dean.menezes
Posts: 135
Joined: Sat Nov 15, 2008 3:47 am UTC

### Re: Really basic stuff that was never proven

Lemma 1

Let [imath]p[/imath] be prime and a positive [imath]<p[/imath]. Then no positive number [imath]b[/imath] can be found less than [imath]p[/imath] such that [imath]ab \equiv 0 \pmod{p}[/imath].

Proof

If the theorem is false, then we have numbers [imath]b, c, d, \ldots[/imath] all [imath]<p[/imath] such that [imath]ab \equiv 0[/imath], [imath]ac \equiv 0[/imath],

Let [imath]b[/imath] be the smallest of all of these so that no number less than [imath]b[/imath] has this property. Clearly [imath]b>1[/imath], since otherwise [imath]ab = a < p[/imath]. Since [imath]p[/imath] is prime, it cannot be divided by b, but lies between two successive multiples of [imath]b[/imath]: [imath]mb[/imath] and [imath](m+1)b[/imath]. Let [imath]p - mb = b'[/imath] so that [imath]b'[/imath] is a positive number [imath]<b[/imath]. Since [imath]ab \equiv 0[/imath], [imath]mab \equiv 0[/imath] and so [imath]a(p - mb) = ab' \equiv 0[/imath]. This implies that [imath]b'[/imath] is one of the numbers [imath]b, c, d, \ldots[/imath] and it is smaller than the smallest of them QEA.

Theorem

If neither [imath]a[/imath] nor [imath]b[/imath] is divisible by [imath]p[/imath], then [imath]ab[/imath] is not divisible by p.

Proof

Let [imath]\alpha, \beta[/imath] be the least positive residues of [imath]a[/imath] and [imath]b[/imath]. By hypothesis, neither of them will be zero. Now if [imath]ab \equiv 0 \pmod{p}[/imath] then [imath]\alpha \beta \equiv 0[/imath] which contradicts the previous theorem.

The fundamental theorem of arithmetic follows by induction.

antonfire
Posts: 1772
Joined: Thu Apr 05, 2007 7:31 pm UTC

### Re: Really basic stuff that was never proven

That's just a (not terribly nice) way to redo two sentences in the original proof.

Yes, it's "the important" lemma.

Yes, all the nasty shit in the original proof is mostly to formalize just what "unique prime factorization" even means (something you didn't bother doing at all).

No, what you wrote is certainly not "truly trivial". It is, I would say, slightly more complex than the proof sketched out originally. It just brushes the boring details under the carpet.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

Spambot5546
Posts: 1466
Joined: Thu Apr 29, 2010 7:34 pm UTC

### Re: Really basic stuff that was never proven

So i've been dicking around with some math stuff and i was looking at just why a negative * negative = positive, and negative * positive = negative.

I know that they do, and can prove it algebraically, but not when looking at the actual numbers.

Spoiler of algebraic proof
Spoiler:
$a + (-a) = 0$
$[a +(-a)]b = 0 \ast b$
$ab + (-a)b = 0$
Thus
$(-a)b = -(ab)$

But multiplication is really repetitive addition. a * b is a added to itself b times.
a + a +a...
-a*b is -a added to itself b times.
(-a) + (-a) + (-a)...
a * -b is a subtracted from itself b times
a - a - a...
and -a * -b is -a subtracted from itself b times
(-a)-(-a)-(-a)...

But those last two are wrong. They should be
-a - a - a... and -(-a)-(-a)-(-a)... respectively.
But i can't find any reason to justify why (besides, i guess, the fact that that matches the algebraic proof). If a * -b is a subtracted from itself b times then why must i start with -a? It doesn't seem to make sense. :-/
"It is bitter – bitter", he answered,
"But I like it
Because it is bitter,
And because it is my heart."

Dason
Posts: 1311
Joined: Wed Dec 02, 2009 7:06 am UTC
Location: ~/

### Re: Really basic stuff that was never proven

Spambot5546 wrote:But multiplication is really repetitive addition. a * b is a added to itself b times.
a + a +a...
-a*b is -a added to itself b times.
(-a) + (-a) + (-a)...

ok
a * -b is a subtracted from itself b times
a - a - a...

Clearly this is wrong. Take a *(-1)
a - a = 0.
So we already knew this was wrong. But this just proves the point anymore. Your intuition is failing you here. What if we redefine your way of doing things as:
a*b is 'a' added to 0 'b' times. Notice that the a*b as "a added to itself b times" isn't really right because then 2*3 should be 2+(2+2+2).
Using the new method then we can consider a*(-b) to be 'a' subtracted from 0 'b' times. Then we also get (-a)*(-b) is (-a) subtracted from 0 b times. Play around with it.
double epsilon = -.0000001;

Spambot5546
Posts: 1466
Joined: Thu Apr 29, 2010 7:34 pm UTC

### Re: Really basic stuff that was never proven

Dason wrote:What if we redefine your way of doing things as:
a*b is 'a' added to 0 'b' times.

That right there is exactly the kind of redefinition i was looking for.

With that a * -b would be 0 minus a series of "a"s and the whole thing works out.
"It is bitter – bitter", he answered,
"But I like it
Because it is bitter,
And because it is my heart."

Syrin
Posts: 290
Joined: Thu May 24, 2007 7:10 pm UTC

### Re: Really basic stuff that was never proven

Let a,b be be elements of a ring R. Let "-" denote additive inverse - that is, -a is the additive inverse of a, so a + -a = 0, where 0 is the additive identity of R. Then, (-a)*b = -(a*b), that is, a*b + (-a)*b = 0. This is a rather trivial consequence of the distributive property (and also that 0*x = 0 for all x, which is perhaps not immediately obvious), and so holds for any ring. It just remains to be seen what "negative" might actually mean in some cases.

antonfire wrote:That's just a (not terribly nice) way to redo two sentences in the original proof.

Yes, it's "the important" lemma.

Yes, all the nasty shit in the original proof is mostly to formalize just what "unique prime factorization" even means (something you didn't bother doing at all).

No, what you wrote is certainly not "truly trivial". It is, I would say, slightly more complex than the proof sketched out originally. It just brushes the boring details under the carpet.

Note: the person who made the proof is not the person who said that there was a trivial proof. In case you maybe wanted to bottle up your vitriol for another time.

Quaternia
Posts: 218
Joined: Mon Jun 15, 2009 6:18 pm UTC

### Re: Really basic stuff that was never proven

antonfire wrote:Give one.

It's far from a trivial proof, but there is an elementary proof that shows that the sequence P consisting of unity and all the prime numbers is a basis of the sequence of natural numbers. "Three Pearls of Number Theory" by A. Y. Khinchin talks about it. The proof is by L. G. Schnirelmann.
(the fact this was always assumed in high school and first year university also hurt my heart, but I guessed it was just too hard to be covered).
Yakk wrote:hey look, the algorithm is a FSM. Thus, by his noodly appendage, QED

Kurushimi
Posts: 841
Joined: Thu Oct 02, 2008 12:06 am UTC

### Re: Really basic stuff that was never proven

I've been trying to prove that that for an ellipse with a semimajor axis of a and eccentricity of e the distance between the center and the focus is ea and the distance between the directrix and center is a/e but I'm kinda having trouble. Any hints on what path I should take?

benhowt
Posts: 4
Joined: Sun Sep 19, 2010 11:21 am UTC
Location: London

### Re: Really basic stuff that was never proven

WarDaft wrote:Very rarely (I would venture perhaps even never) has the proof been given that 1+1 does in fact equal 2 before more advanced topics are covered.

Refer you to Bertrand Russell's Principia Mathematica, volume II, page 87. I haven't read the proof, but suffice to say that by the time it is done you're at page 200 and something. Also note that many other laws are discussed previously (in Volume I)

Eebster the Great
Posts: 3484
Joined: Mon Nov 10, 2008 12:58 am UTC
Location: Cleveland, Ohio

### Re: Really basic stuff that was never proven

WarDaft wrote:Very rarely (I would venture perhaps even never) has the proof been given that 1+1 does in fact equal 2 before more advanced topics are covered.

Isn't that essentially the definition of 2, though? I mean 2 is usually defined as something like the successor of one (let's write it S(1)) or as {0,1} = {{},{{}}}, which are essentially equivalent. Given 2 = S(1) and 1 = S(0) and a + S(b) = S(a + b) and a + 0 = a, we know 1 + 1 = 1 + S(0) (by definition of 1) = S(1 + 0) (by definition of addition) = S(1) (by definition of addition) = 2 (by definition of 2).

So that one truly is a trivial proof.

Spambot5546
Posts: 1466
Joined: Thu Apr 29, 2010 7:34 pm UTC

### Re: Really basic stuff that was never proven

Yeah, at that point you might as well ask for a proof that 1 = 1. That addition functions the way it does on our fingers, and things like the properties of equality, have to be accepted a priori because there's really no way to prove them and nothing in math works without them.

To put it another way, hold up one (1) finger. Now hold up one (1) other finger. Note that you have held up two sets of one (1) finger. Careful observation and measurement will demonstrate that you are now holding up two (2) fingers. Ergo 1 + 1 = 2. QED.
"It is bitter – bitter", he answered,
"But I like it
Because it is bitter,
And because it is my heart."

Eebster the Great
Posts: 3484
Joined: Mon Nov 10, 2008 12:58 am UTC
Location: Cleveland, Ohio

### Re: Really basic stuff that was never proven

Spambot5546 wrote:Yeah, at that point you might as well ask for a proof that 1 = 1. That addition functions the way it does on our fingers, and things like the properties of equality, have to be accepted a priori because there's really no way to prove them and nothing in math works without them.

To put it another way, hold up one (1) finger. Now hold up one (1) other finger. Note that you have held up two sets of one (1) finger. Careful observation and measurement will demonstrate that you are now holding up two (2) fingers. Ergo 1 + 1 = 2. QED.

Technically once you bring fingers into the picture, you are talking about physics, not pure math. However, if we assume that we can model your process of finger-addition or whatever as an operation, we could easily prove that there is an isomorphism between the group of natural numbers with addition and the group of fingers with finger-addition.

gmalivuk
GNU Terry Pratchett
Posts: 26822
Joined: Wed Feb 28, 2007 6:02 pm UTC
Location: Here and There
Contact:

### Re: Really basic stuff that was never proven

Any "proof" that 1+1=2 isn't done because we have any doubt that 1+1=2, but rather because we need to make sure that the system we're building implies the mathematical results we expect. If Russell's proof ended up showing 1+1=3, of course it wouldn't mean that we'd been wrong all these years. It would just mean that whatever Russell started with turned out not to be the correct way to build math on solid logical foundations.
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
---
If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome

(he/him/his)

voidPtr
Posts: 140
Joined: Sun Apr 26, 2009 6:53 pm UTC

### Re: Really basic stuff that was never proven

One thng I don't ever remember being proved in grade-school maths is that the angles in every triangle add up to 180 degrees.

Actually, I remember a lot more of what wasn't proved than what actually was proved, some of it for good reason, some not.