Wednesday, December 26, 2007

DEDICATED TO SPACE SHUTTLE COLUMBIA

Chapter 4. MATHEMATICIA.DEDICATED TO SPACE SHUTTLE COLUMBIA.

What am I going to teach about mathematics.
Only what I know that is grade ten mathematics.
And I have forgotten most of that.
So we will make it second grade mathematics.
That is the simplist mathematics that I can come up with.
But what can I prove with simple mathematics.
I think I will sent out to prove Einsteins Unified Field theory.
That is everything can be put into one equation.
And that is a task.I have twenty pages to explain what simple mathematics is all about.
Then in the remaining ten pages I will explain Einsteins unified field theory.
Since it doesn’t exist how can I explain it.
The truth is it does exist but it is not believed.
And if I understood it I wouldn’t be able to explain it.
But since I know nothing about it.Except that it unifies the four forces.
I will write Stephen Jeffreys unified field theory explained in terms of simple math.
Why so simple.? One of the rules of science is KISS.
Keep it simple Stupid.
And it means simplify the problem and that is what I will be doing.
I will be simplifying.It is up to university to coimplicate the problem.
Because before Iam finished I will have an experiment.
Because a theory is of no value without an experiment.
Everything I do is done to show the right way to do science.

Iam not unscientific in my method.
Because to start off with you have to make an educated guess.
And thats what I will do.
What is 3+3=?
My educated guess is 3
3+3=3
Because my educated guess can be proven right.
And if it could be then it would be an experiment.
And the experiment is add three threes. 3+3+3=?
And the answer is nine.
And that prove me right.
That three threes are nine.
Because that is why I said that 3+3=3
Because two is not three.
And it really isn’t.
Unless you use six.And that is exactly it.If you use six.Then it is 2+3.
And that is what is really is not.It is 2*3.
And that means 2+3= 2*3 Because that would mean addition and multipication were
the same.
And that is a rule.2+2=4.
And it means addition and multiplication are always the same.
And if it does then perhaps 2+3=5=2*3
And that is the alternative.
But we will choose to make multiplication the correct answer.
That means 2+3=2*3=6
Now we can see how 3+3=1/3
That means a third of nine.
Which is totally right.Three twos are equal to one two.
And that isn’t what we have said.
We have said two twos are equal to three.
And five isn’t three twos.
3+2=5 But if you add two fives togther.You get 5+5=10
Then you take 1/3.
And it is equal to 3.333’
That means 1/3= 3/3
And it means 2/3=1/3
And that is confusing.because 3 is really 1/3.because the answer isn’t nine.
It is ten.
So 3.333’+3.333=6.666’
And that isn’t what we have said we have said it’s 3.333’
And that isn’t proof.
And it really isn’t.
I have set out to prove 10 is the answer.
And I have failed.

Because you cannot say 3+3=3
Not unless 2+3=6.
Because the two are not the same. 3*3=9
So 3+3=9 And that is right.
Because that is what I set out to prove.That 1/3=3/3
And I have succeeded.
Becasue that is not what I believe.
Unless I can understand why it is so.
But I will explain what I mean.
There is a reason why nine isn’t right.
It isn’t the right value for 9.
Because the right value for nine is 9.999.
And that is ten.
So if we make 3.333’into ten.
How do we round it off.Since .333 is infinity.And 1/3=3/3 is our rule.
Then it is four.
But four goes into ten 2.5.
And that is infinity.
2.5.
Because it is rounding off 3.333
Of course we could round it off to three.
And that wouldn’t be infinity.
And there is no other way to round it off.
Except to make it 6.
And that is an answer we could say .333=.666
Then 3.666+3.666+3.666=10.
And it actually equals 11.That is 10.999999
But that isn’t really 11.
It could be called 10.
But it isn’t.It is 11.

So if we round off three lots of 3.333+3.333+3.33
We are adding something and that is a ?
Because 1/3 of ?= what we add to .333333.To make it .4
Because 3.4+3.4+3.4= 10.2
And that is what ten really is.
And it may be right.That ten can’t be two things.
Because ten is the basic number.
If there is more than one ten.Then there are three ways of doing things.
And there really are .
Three tens.
That is 9.999,10.999’,10.2,10
That is four but really ten does exist.because it is rounding off 3/3?.
If we add an infinite one to an infinite nine.We get 10.11111
We don’t get ten.
That means we can’t add infinity.
Otherwise we would be able to add 1 to 9 and get 10.
This is really interesting.

Because people wouldn’t know what the third ten is.
They wouldn’t know that it’s 10.
Because you can round it off to ten.
And that may not be possible.
There may be nothing you can add to 9.999 to get 10.
Say you take it to three decimal places 9.999+.111= 10.11
And that is really two decimal places.
And if you take it to only one place.You get 10.
9.9+.1=10
Anything else is another figure.
That means to get a right answer for ten you need to take it to only one place.
That is 3.3+3.3+3.3=10 And it doesn’t.It equals 9.9.But you can add 1.1 and get 10.
And there is a right number of ones to add to ten.
So you get ten and for 3.33.That is infinity.Because you can add .1 to infinity and get 10.
And you get 1. 1.0999999
So really that 2.
And that is totally what I want to prove.
I want to prove 10=2
And there is a reason and it is that infinity and finity are two seperate things.
You can’t just add infinity to infinity and get 10.
That is you can’t add 9.99999’+.1111111=10 And get ten.
But you do there is a figure that is right.And that is 10.11111
And you can round that off to ten.
Because it is only .1
Just like you can round .99 off to ten.
It is 10.109999
Nearly 10.2.
Thats what you get when you make 3.333=3.4
Because that is the right figure.
And it isn’t.Because it is infinity.If we want it rounded off then 10.2
Is the right figure

And we have really proved something about 2+3=2*3
Therefore 3+3=3*3
And both can’t be right.But the rule is consistant.
If we let 2+3=2*3
Then 3+3=3*3
That means 5=6
And 6=9
That means 5=9
And it really does.

That is why Iam writing mathematica.Because it could be Maths.
And thats all it is.
It is a book of mathematics.
To explain everything.
That is simple.Like unified field theory.Because that is really simple.
And I believe so.
That if you know the answers.
It is easy to ask the questions.
If 5=9 Then two fives=9.999
And that isn’t what the equation sais two fives must equal 18.
That is two fives must equal three sixs.
And that is good mathematics.
Because ten is not infinity.
It is 2.5 lots of 3.4.
Or three lots of 3.333’
Or two lots of five.
And that is the way to look at it.That is 5=6.That is 2+3=2*3
Then 10 is two lots of six.
That means addition is right.And multiplcation is wrong.
2+3=5=2*3
That means addition and multiplication are the same for five.
And that is a rule we could use.
Because there is no law against making up rules.
As all of the rules of maths are made up.
But they have to be consistant.
That means two fives are ten.
4+6=10 4*6 also has to equal 10.
And it does according to our rule since twice five is ten.
24 doesn’t enter into it.
Because it is not twenty four.

What we have to understand is how 4*6=10
Because the rule is 2+2=4.
That all multiplication and addition for two is the same.
But the way we are using it.
All multiplication and addition are the same.
And there is nothing to stop us using the rule for two for one.
Because if we use it for two 20=2*10 10+10
And that isn’t right.because twenty is four fives.
It is 5+5+5+5=20
So there are same number added as there are mulitpled.
2+3=(2*3)*5= 20
And that proves the rule.The rule is right for two.
And it is right for one.
As long as the rule.
Is 2=1
And that is as close as I will go to proving Einsteins unified field theory.

Because it really is all about two=1
Because more than one thing can be one.
And that just isn’t true of four.
Four things can’t be one.
And that is what Iam asked to prove that 4=1
And so I will set out to prove just the opposite.
That four does not equal one.
And 2+2=4 Proves you can have one equation.
That is two things.And it is two lots of two.
So it is really four things.
But you can’t start with E=MCsquared and prove 2+2=4.
And that is what Iam all about.
Proving E=MCSquared is not two but three.
And the four forces are not 2*4=8
But 3.333+3.333+3.333=9.999
But if 2+2=4 is right.Then the answer is 10.2.
That is rounding off 3.333 to 3.4.
And it makes perfect sense to believe there are really five forces.
5*2
Giving a total of 10.2 forces.
Now how can you have .2 of a force.
You just can.Because that is the prediction.
If 2+2=4 is right then there is such a thing as .2 of a force.

So you can add E=MCsquared in twos.
You can add five lots of two.
Thats five equations for everything.
And that you expess Einsteins unified field theory.
As finite equations.
Not as infinite equations.
Because to do that you need three.
Because you have to make ten.
And you just do not have to.You can leave the figure as infinity.
That is 3 1/3+ 3 1/3 + 3 1/3= 9 3/3
Because everybody except s that 3/3=1
But it really is equivalent to 9.999
And it that difference that make Einstein right.
Because you don’t have exactly ten forces.
Then are nine and infinity.
That makes ten forces.
So the ten force is infinity
What the hell is negative two.?
That exactly it.
There is such a thing as negative two.And it is negative infinity.
Because it really is not.
It is just plain old negative two
But when it is infinity it can negative or positive
And that the hard part to understand.
That there can be an infinity that isn‘t rational.
That is one that is two.
because to make infinity out of a half really is impossible.
Since you would go on making one less.
And that isn‘t sensible.
Infinity is one more.
And thats what a 1/3 is it is one more.
But 2/3 is the opposite it is a fraction that appears to be infinity.
And it isn‘t like a half.
There is always six more.
And that is just it.There is always another two.
And that isn‘t right there are always three more twos.
And if there were two more twos.
It would be infinity.
And it would not.
There has to be one more.
That is three for it to be infinity.
Otherwise it is a rational number.
We will go on to prove that.
Ten is made up up 2/3+2/3+2/3+2/3+2/3= 3.333‘
Now this is the most unexpected thing of all.
Because five twos are ten.
But ten divided by three.And it is 3.333’
And that makes five twos something that would appear to be right.
Because 15 twos make up ten.
2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3
Now is that right.
Since it it is two it is right.Because it is also three.And it does appear that way.

That five 2/3’s is not a rational number it is infinity.
And that is a question about 1/3.
Is 3.333’ infinity.
Or is .333’ infinity.They are really the same.
And they are just the decimal point has been changed.
What does it mean that 15 twos is ten.
It means thirty twos is twenty.
But hold on 2 =2/3+2/3+2/3
So five twos are 2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3
And that is exactly right.It answered a question.
15 is five time three.
That means there are five twos in ten.
And that is what an infinite two is.
The number of 2/3 in ten.
And that is five.
And thats right two is really five.
And that doesn’t make much sense.
But when you add three lots of two.
You are adding 3+2
Because you are adding 3*2
And that is just it 3*2=2 And that is an equation.
And it means of course 3*2/3=2
But what about ten 15*2/3=10
And that is 3*5*2/3=10
That means we have fifteen equations.And we really have five lots of three.
Because we have two equations that are added up to six.
And that is five lots of two.
Which suprisingly doesn’t give you ten but 9.9999’
And that is just perfect.
Five twos are 9.999’
And that is only true if we are adding two thirds.
That makes two infinity.
And it really does.
Because 3 lots of 3.333’ are 9.999.
That means you will get the same answer if you use ten.
As you would with 9.999.
Because you won’t.
Because you could argue that 9.999 is not ten.
And that five twos are ten.
And that is a perfectly sound arguement.
And it really is what it means to not believe.
Because it proves that 666 is really finite.
And it does prove that 666 is not.

That it is infinite.
But what about ten.If 666 is ten then it finite.
Then 2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3+2/3
=10
That means 18 has to rounded off to twenty.And that can be done.
The infinity is finite.
And this was not what the co-author expected.
He expected that .666 was finite and .999 was infinite.
But he didn’t understand that eight is not ten.There are twelve 2/3’s in eight.
And that means twos sixs.
And ten is the opposite to 666.
3+3=6.
That is why two threes are six.
Because when you add 3 to 3 you get six.
That means two infnities is always six.
Becasue infnity is 3.333’ and it if it is 10/3 then it is.
But if it is 1/3 then .333.
But you when you multiply it by two it 6.666
And that is what we are doing when we use 2/3.
We are saying infinity is two.
But is light really two.
That is what building the telescope shall confirm.
If light is really two then the picture you get when two is five.
Will be terrific.
But will it be as good as using mirrors that are 1/3.
Because it may not be.
But I suspect that 5/3 will produce a better picture.
And I mean on your wall screen 3DTV
.Becausue 1/3+1/3+1/3+1/3+1/3=5/3
And that is not two.
So it doesn’t add up to.3.333’
And it is 1.6666665.
And if multiply that by fifteen..4.9999995
And the answer is 9.999999.
And that proves that 3 can be five.
Just the same as two can.
So if you have both you have binocular vision.
You have one lense of five that is five 2/3.
And another lense of five that is is five 1/3.
That means you can show pictures faster than light.And it doesn’t.
Because we can’t accelerate light.
And we can if we slow it down.
But we can’t accelerate it faster than light.
And maybe we could.
Get light to travel at 3C.
Because light were accelerating three times faster.
We would see two pictures.
And we would.
But they would appear to be one.
Because light doesn’t need to travel at 2C.
It can travel at the speed of light.
And it can travel slower,.Because if we reflect the two images between two mirrors.
We can choose the speed we want the image at.
Because each and every image that you divide light into has a different speed.
And that is the theory.
But it is a good way to test it.
Because if light is travelling at 1/3.
Then it’s speed is given by this equation 1/3E=1/3M+1/3C+1/3C
That is a third energy.
And you you tell how much energy each image is.
And when it is 1/3 energy it is 2/3 light.That is the speed of light.
2/3 of 186,000 miles per second.
And if that equation is right.
Light will be right for two images.
Because it will be 2/3+2/3
And that is right it will be 4/3.So it will be 666.
And that is what I want to prove.
That TV should be two dimensional.
That if three and two are put together.
Then you get six.
But if we use addition you get five.
That is why you need two speeds.You need an image that has a speed of 1/3.
And an image that has a speed of 2/3.
Then when you put them togther you have the speed of light.
Now this is an experiment in televsion.I won’t guarantee that by slowing the images down.
And putting them togther.
You will get one speed.
And I will guarantee it.If the hypothesis that a mirror can slow light down is true
Speed of images is what it is all about.
Because you have to see both images at a different speed.
And when the speed is right for the image.
The illusion of 3D is not really an illusion.
It really is 3D.
So we need cameras.
And we have one that is two.It is three cameras in one with five lenses of 2/3.
And we have another camera that is three.It is six cameras of five 1/3’s.
And that is our two images.
You would think that 1/3 is wrong because you need
a lot of lenses.
But you really don’t need that many.
Because you need twice as many as do for two.
And you only need three.
And you really do not.because we have divided distance into two.
That is what ten really is.
When we express two and three as five.
We are adding two pictures togther.
To make ten.
And that is the secret.Because when we have six fives.We have 30.
And that is 9.999’
So it is the same as two you just need six lots of five lenses of 1/3.
And that sounds difficult.
And it isn’t easy.
Because it proves me right.
That two and three and right togther.
Because I must believe that.
And I just do not.
But the eye is a steroscope.And that is the way it produces three D.
By two images.
They are put together to make four.
And they are not.
Because the eye would not be designed that way.
But it it is the eye which saw the tree of knowledge.
And desired the fruit.
It is the eye that leads to temptation.
So it is right that the eye should mean 666.
And the problem of 3D TV will not be solved without comprehending.
How wrong six is.
Because we have choosen infinity.
To be our TV.
It is will be a good one.
We could have choosen 10.
That would mean lenses that are not infinity but are exactly two.And there would be five of them.
And the other camera would not be three because thats infinity.
It could 3.4 because that is infinity rounded off.
And that would work.
But it wouldn’t be infinite.
The same goes for using two cameras with five lenses of exactly two.
They add up to ten.
And if the measurements are the same.
The lenses will be two.And they will add up to ten..That is how we know they are two.
That is a camera that is not exactly infinity.
And it is perfectly sound.
For television.
Because the picture may be even better.
But it will be saying that two is not infinite.
And that is the right thing.
Because two is not.
But distance is infinite.
And the distance of the futherest star can be computed by using the size of the universe.
And letting that equal ten.
Then you can use 10E=10M*10C*10C
And that is our equation for distance.
Figure out how many kilometers the equation says the universe is.
That is a hard problem.
But it will be solved someday.
And the distance of the universe is what we need for TV.
That is we have to reflect light between two mirrors to infinity.
So that energy approaches zero.
And when it does energy is approaching 10.
That is energy has travelled from infinite levels to zero.
Our focus has to be the distance light travels when it travels to infinity.
And that is 1/3.
Because the equation means that 2/3MC+1/3MC=3/3MC
Which means 1/3=MC
That is when you use addition.
And it is no different when you use multiplication.
1/3=2/3MC*1/3MC
1/3=.222221 MC
And it is different.That is because we have changed the rules.
1/3*1/3= 3/3
But when we add we can put 2/3 together and add 1/3.
But if we could do that when we multiply then we would get an infinite two.
And that is what we have got.
Because we just changed the rules.
When you add 2/3 to 1/3 you can also multiply 2/3*1/3.
That means addition and multiplication are same for two and three.
That is when they are added.
And the same goes for 2.When you add 2 you add 2/3.
Then you get 2/3+2/3=4/3
And if you have three thirds.
You have 2/3+1/3=3/3
But when you multiply 2/3 togther you get 4/3.
But when you multiply 1/3*1/3=3/3 you get 3/3
So the rule is quite simple 3/3=1/3.
Becasue 1*1=1
So 1/3*1/3=1/3.
And that is proof of the idea that 1/3=3/3.
And it isn’t clear what happend when you add 2/3+1/3
Because when you multiply you have to get two.
Because 2*1=2.But if addition and multiplication are the same.
Then the answer is 1/3.
Because it is 3/3.
And they are equal.
So that confusing.
Because it isn’t the right way to do it.
But if we could do that.
Then we could have equations that balance.
For two and three.
Because they would have to balance for two or for three.
That is 1/3+2/3=3/3
And if we balance it for two that means 2+2=4
That means 1/3*2/3=3/3
And that is totally right.
Only the real answer is an infinite two.Which is not three.
And that is why we balance two equations.
By putting two together.
We put two equation together in twos.
And that is as simple as it can be.
You don’t want to believe in 2+2=4
But it is made so easy.Even though you can show four is an infinite six.
Still you think there is nothing the matter.
Because three and two and not meant be be one.
When they are multipled and not added they are one.
And the rules do not apply.
Because all of these rules I have been talking about apply to six.
That is 6+6=12
And 6*6= 36
But if we make multiplcation and addition the same.
We have to decide what is right for two.And for two addition is the same a smultiplcation.
So we can adjust multiplication to be the same as addition.
That means 6*6=12
And that is a new rule.
Always make addition right never multiplication.
And the right answer is to make multiplication right.
That means 6+6=36
And that wouldn’t make sense.
But it makes just as much sense to let three be right.
Then 3+3+3=9
Or 3+3=6
That means 6+6=12.
And that is correct so addition is the same as three.
That gives us a clue multiplication may be the same as two.
That means 6*6=12
And that makes perfect sense.3 is for addition and 2 is for multiplication.
But if multiplication is the same as three.
Then 3*3=9
And 6*6=18
And that is more what I believe it to be.Because three sixs are 18.
So we need to make addition the same.
And it is.Because 6+6+6=18.
And that is our new rule 6 is multipled to give 18.And 2 18’s are 36.
That a total of six sixs.
And that is no problem because the rule is for three.
If we have three sixs they add up to 18.
So we can see that the rule is the same as for nine.
Unless we want to use two then the rule is 6*6= 12.
And that explains everything.
Because twelve is two sixs.
And when we add them we get 12.
So which is right.Two is right for addition.
When you use two sixs.Then you should add up six in threes.
And that means you add up 6+6+6
And not 6+6
Because the rule for three is right for six.
You can use the rule for two.
And it produces an entirely different result.
You get equations that balance.
And you might.
Because for two equations don’t balance.
E=MCSquared F=MA
Don’t balance for two.
And if right equations don’t balance for two.
Then two isn’t right.
We need to change it to be like three.
3+3+3=9
2+2+2=6
That means 2*2=6
And that is our change to the rules.
Why are we changing the rules.Because this is a book of mathematics.
And we have the right to change the rules.
Because they were made up.
And everybody knows that the enemy doesn’t like rules.
But he will follow them.
To the very letter.
Because they are answers.
If two is three then what is 2+2=4?
It is clear that 2+4=6
And that is the rule.That must be followed.
But the other rule is 2+2=6
Because it is for three and it 2+2+2=6
And you would almost believe multiplciation and addition were not the same thing.
And they are.
But only for two.
That is why we can use the rules to prove 18 right.
Because that is what people who are atheists will say.
They will say 6*6= 18.
Because the rule isn’t right for six.
But it’s either that or 6*6.Which is of course 36.
But if you add three sixs together.
Then two sixs multiplied should equal 18.
That is only if three is right.
And that is the answer two is right
2*4=8 Because that means infinity.
And it doesn’t.It is only eight and nine is infinity.
And if you want to say 18 is both two and three.
You have to add two togther 2+2=6
That means when you add three sixs.You multiply 2*3
And you get 6+6=12
And that rule is right for two sixs.
But if we have three sixs we have to use a rule for three.
And that is as simple as it can be.
It depends on how many sixs you have.
And we have proven the number to be something that can be calculated.
That means people can add equations 2+2=6
And it is totally right Since 2/3+2/3+2/3= 1.9999998
And infnite two.
And that is 2+2=6 means.
It means two infinite twos are equal to six.
Six lots of..666
We are looking for rules so we have them.
When we have three sixs we use the rule for nine.
When we have two sixs we use the rule for two.
That is when we are adding two sixs.
We make the rule 2+2=6
Then 6*6=18.
And that is confirmation that no matter how we do it.
Three sixs is eighteen.
And that means we add equations in sixs.
We add 2+2=6+
2+2=6+
2+2=6
6+6=18

And we get 6+6=18 And that makes addition the same as multiplication.
But it is reversed.
And that is what the number is all about.
It is about reversing.
You can’t change the rules without changing everything.
And when you change 2+2=4.
You also change 6+6.
The rule is consistant.
But not the same as the rule for three.
If we want to build a quantum computer.We need a different theory.
because this theory is that Einstein is right.
And a unified field theory is possible.
But we build a relativity computer.
One that works on the idea that 2+2=4 will get the right answer.
And not when it is finite but when it is infinite.
And it doesn’t seem right.
But as we have just explained.15*2/3=9.99999’
2/3=.666’
2/3 is not the same as two.It is really an infinite two.And if that is true.
What about 3*2/3.That is really two.
Because it is exact and it is not.It is nearly two.
So there are two twos.
And we can add them together to get four.
.666+ 1.998=2.664
And that is a four that is three.And it doesn’t make sense for four to equal three.
And it makes perfect sense when you are using 15.
One is .666.Two is 1.332. And three is 1.998.And four is 2.664
And that is totally the opposite of right.
Because there ahs tobe two infinite twos.
because it isn’t one.
So when you add them togther you get four.
That means one is.666.
And you can’t treat two as one.
because an equation that is .666 is an infinite two.
And it really is 1.
That is what 2+2=4 sais.
So we will get back to fifteen.
We have fifteen lots of 1.
And that is what ten is.
It is a new system .10 is really 30.
And it isn’t it is fifteen lots of 2/3.
So if we make a computer that is 2/3
Then we have to use .666
That is we have to have three lots of two.Three lots of .222
That is two infinite ones .1111*2
So how do we make off and one infinite.?
That is the question.And the answer is that off is infinity and that one is infinity.
We simply use infinity for off and on.
And that doesn’t really answer the question why is it so.?
And it most certainly does.
Because light and darkness must both be infinite.
And that of course is false.
Light is infinite and darkness is a lack of light.
But off is the opposite of on.
So we can say that there is no current.
And that infnity is not negative.
But positive.
So we will begin with three lot of two.
Because the real infinite two is .2222
And we have fifteen lots of .666.
Which is 9.9999’
That means we have five twos.
And we really do have five lots of .666+.666+.666
Only now we have three infinite twos.And that is right for six.
There .222’ there is .666 and there is 1.9999998
If we add them together we should get infinite six.
We get 2.8888886
That means 3=6.
And that is obviously false.
So there is something very false about using .666 as if it were two.
But when we use fifteen we make two an infinite three.
And that what we have to do to get an infinite 3D TV.
Because we need to use 3D glasses.
And so we have to divide the image into two.
But when we use fifteen images of two.
We get a 3D picture.
But what about a computer.
Is it the same.
There are 45 lots of two.Because there are fifteen lots of .666.
Which divides into three .2222
And we are right to use .2222 as infinity.
Because it is false.
But it adds up.
To produce the infinity that we believe in.
So maybe the picture isn’t true infinity.Because it is made up of infinite twos.
Will make no difference.
And we can expect that there will be a difference.
Because to the eye it will be infinity.
But to the heart it will be both three and two.
That means added togther.
And it really does.because if we multiplied.It would be the same as adding.
Now I must be leading you astray.
And Iam leading myself astray.
Because I really wanted a quantum computer.
But I didn’t want to use three.
That is there is off there is on and there is nothing in between.
And that is the kind of computer that I have built.
One that is true false.
Because it is two ones.
And not one and zero.
And one can either on or off.
And that is completely OK.
It means there are four positions 11 ON OFF 11 OFF ON 11 ON ON 11 OFF OFf
And it means power on and power off.
But since you don’t know which infinity means power on and power off.You have to make one negative.
That means -.11111’
Is power off +.11111 is power on
And there may be no such thing as negative infinity.
And there really may be.
So fifteen twos become one 10.
And that means the computer can use base 10.
It just has toconvert it to infinity base 2.
10=15*.666
And that is the equation that makes this computer work.
But it won’t gie you true false answers about equations.
And it really will not.
Since true is .9999
And false is .66666
Three of them together which is two is false.
That is the value for two that is not infinite.
And we have fifteen two together that is true.
And that would be fifteen 11
And that is right.But because it is zero doesn’t change the fact that it is 1.
And that the difference with this computer.There is a third alternative.
And that is zero.
You can have 01 00 11 10
And that is just like a normal computer.
Except there are fifteen lot of .666
So there are 45 lots of two.
Which is nine lots of five.
Anyhow I don’t think this particular computer is the best we can do.
Because it has the same capacity as any other computer.
And it really doesn’t.
It has an infinite capacity.
Because that is not negative it is postive.
Because it adds up to infinity.
You can use infinity instead of ten.
And it makes a diffference when you use infinity.
And it might make a difference when you want to understand the truth.
Because two can be added up to infinity.
Does mean mean that .666 is right for two.
That means we need a computer that is not .666
One where two is the correct value.
And since we know that .222 is really infinite two.
Why not have five .222.
And that give you 1.11111
And that is as good as 10.
It means a value for infinity that is not the same as .333+.333+.333
And it really is not the same.
Because one is a value for fifteen.
And one is a value for ten.
If we make .222 a value for fifteen.We get 3.3333”
And that is totally right.It proves 15 to be the magic number.
So we build our computer based on 1/3.
And we add three fifteens togther.And we get 45.
Which is nine fives.And since there are five lots of .222.
There are nine lots of 1.1111’
That is the way to build our quantum computer that one that proves
666 wrong.
The Unified Feild theory mathematics project.
What is it it is to add up every equation in the world that could apply to Einsteins Field theory.
Adn how do we do this.The same way as we have said.
By using .2222 as infinity.
Because that is right for 2+2=4
.2222+.2222=.4444’
But what is wrong with it for multiplication.
The answer is infinity 0.0493827
And this really means four and it must.
So if we want an infinite value for four we multiply.
Now to get three we multiply by 15.
So we will multiply .2222*.2222=.0493827*15=.7407405
And unlike pi we know that it works out to three.
It is just an educated guess.
Unless the rules don’t apply to 2+2=4
That is multiplied it is something else.
I will leave you to investigate what that something else might be.
And I will go to to the supercomputer.
Having build the computer that uses 15.
We have a means of changing two to three.
But if we want a superequation that is both two and three.
And it really two.
Then we can have one.
By division.By dividing the superequation by 15.
That will give us two.That is if we add equations in threes.
And get the ulimate superequation.
We change that to infinite two by simply dividing by 15.
So we can compare the superequations.
They will be different.
The one for two will not be the same as the one for three.
And as our computer automatically assumes two is false.
It may have a breakdown.And it might because it will assume division by two when the object is three is false.
And it will assume multiplication by 15 when they object is two is true.
And that is the question is it true that 2+2=4*15= 1/3+1/3+1/3=99.999
No it is not true.Since four is 3.333+3.333=6.666
When we double it we would would expect double .6666
And that is what we get 13.3333’.
We actually have to multiply by fifteen again.
That is we multiply by 15 squared.
2+2=4*15*15=1/3+1/3+1/3=99.999
And that can be expresses as E=MCsquared.
2E=2M+2C*C=1/3E=1/3M*1/3C*1/3C=9.999MC
And that is an important equation.It prove C squared is 15.
And it might prove 2Csquared is multiplied by fifteen
To get 1/3 C squared.
And it isn’t multipled by fifteen.it is multipled by 15 squared.
And that gives us a value of 100.
And that is 6.666 works out.Because it is four *15.
That is infinite 4 .44444* 15=6.666’
And it gives us an idea.
Maybe 15 is right for C squared.
When E=MCsquared is two.
Because we make E=MCSquared .2222
.222E=.222M*.222C*C
.222C*C*15= 3.333C*3.333C=9.999C
And that is exactly right you don’t multiply and then muliptly by 15
So it sin’t 15 squared it just plain old fifteen.
And that asnwer .4444C as the speed of light is really.9999
And as long as we use 15 to translate between infinite 2 and infinite 3.
We get the same answer.
Unless we translate three into two.
And have all our equations as two.
Assuming then that two is right and three is wrong.
So if there are any advantages in adding equations in twos.
We can add every single equation that matters.
And we can do the same in threes.
And put the result together as two or three.
We can thus learn everything there is to know.
And that is what I do believe is the whole problem.
Knowing everything is what God does.
And he has shared the way He sees it.
That for Him it is a simple matter of adding everything up.
They you have the equation for everything.
And when you begin to add up.
Begin with 3.
Because it is just as easy to add equations in threes as it is to add them in twos.
And when you use 15 use it to convert 2 to 3.
Not to convert 3 to 2.
Because true should never be converted to false.
And false should be converted to true.
And that is the question you will find impossible to answer.
Why is it so.?
Why is it so easy to convert 2 to 3.
Why is fifteen the magic number.?
15 is 3*5 So it is three.Because it is 5+2
So it is added.And it is right to add.
Because we have proven that addition works just as well as mulitplication.
But if you multiply a superequation that is three by a superequation that is two.
Then multiply by fifteen.
You will get 666.And you will get an infinite two.
Which is what six is.
To finish I would like to speak a word to the the physists and mathematicans at ORU.
Because they will be making the decisions to prove 666 to be be false.
Thus saith the Lord.
To the team at ORU.
Iam your coach.And Iam.I have coached you to win.
By my Holy Spirit.
I expect 110%.
And I know that you will give it.
Because you have all the answers.
They are in the book.
And that isn’t what science is all about.
You have to prove the answers in the book are right.
And to that you will need the best experiment.
And you have one that is almost the best.
Because everybody has a new experiment in slowing down light.
Yours will have to be a special experiment.
One that is easy to perform.
One that can be performed at any university.
And working with what I have given you.
Is easy.
And it might not be.
You might need a new heart and a new mind
To do what you have to do.
And you have just that.
You have a new heart and you have a new mind.
So can understand the things that only the angels understand.
Because they would like to look in at what you are doing.
Because they wonder how God created the universe.
And what the ultimate answers really are.
And they know that having faith is what it is all about.
That without faith you would not be pleased.
By anything God had to say.
You would say it wasn’t good enough.
That the explaination for the creation of the universe was not worthy of God.
But what is worthy is the explaination I have already given.
The one in the book of Genesis.
Because it is the explaination that you are to believe.
Knowing that I could give you a complete matematical expalination.
And that is not beyond my power.
But because I have explained it as if to a child.
You don’t believe.
And I will say to you behold the nail prints in my hands.
And the slash in my side.
And the nail prints in my feet.
Be not unbelieving but believe.
Because Iam He that liveth and was dead and behold Iam alive forevermore.
And I have the keys of hell and of death.
And I have given those keys to you.
As well as the keys to the kingdom of heaven.

Friday, October 19, 2007

SOURCE CODE "DR KARLS MACHINE ROBOT PHYSICS TEACHER

The machine Dr Karl commissioned by saying the fiat words" BUILD THE MACHINE."
Mr. Jeffrey,

As you requested in your last E-Mail,

Dear Gary Rob,
I wnat you to give me the source code for the updated BEYOND 2000 PHYSICS KNOWLEDGE CARDS application.
I want to give the source code to me to give to Dr Karl Kruselnicki for possible supercomputer applications at WASP.
It is a do it yourself kit to program Supercomputers in Western Australia which use my theory...
This is very important Rob I need the source code.
I got burned before with another program for which I didn't get the source code now I can't put it into new languages.
If possible could you also write it in basic when I pay you the rest of the money.
Basic is the all purpose language....... and can be converted into anything else.
That is if you are good with visual basic.
Steve A Jeffrey

Here is the source code for the updates of the Physics Knowledge Cards.



unit uMain;

interface

uses
Windows, Messages, SysUtils, Classes, Graphics, Controls, Forms, Dialogs,
StdCtrls, ExtCtrls, Menus;

type
TForm1 = class(TForm)
Panel1: TPanel;
Panel2: TPanel;
Button1: TButton;
Button2: TButton;
Edit1: TEdit;
Edit2: TEdit;
Image1: TImage;
Label1: TLabel;
Label2: TLabel;
Button3: TButton;
Button4: TButton;
Button5: TButton;
MainMenu1: TMainMenu;
Hlep1: TMenuItem;
contents1: TMenuItem;
Index1: TMenuItem;
N1: TMenuItem;
wwwcjwcom1: TMenuItem;
N2: TMenuItem;
About1: TMenuItem;
File1: TMenuItem;
Contents2: TMenuItem;
Index2: TMenuItem;
N3: TMenuItem;
wwwcjwcom2: TMenuItem;
N4: TMenuItem;
About2: TMenuItem;
procedure Button1Click(Sender: TObject);
procedure Edit2KeyPress(Sender: TObject; var Key: Char);
procedure Button2Click(Sender: TObject);
procedure Button5Click(Sender: TObject);
procedure About1Click(Sender: TObject);
procedure Button3Click(Sender: TObject);
procedure Button4Click(Sender: TObject);
private
{ Private declarations }
public
{ Public declarations }
end;

var
Form1: TForm1;

implementation

uses About, Unit3, Unit4;

{$R *.DFM}

procedure TForm1.Button1Click(Sender: TObject);
begin
Edit1.ReadOnly := false;
Edit2.ReadOnly := false;
Button2.Caption := '&Store Equation in File';
end;


procedure TForm1.Edit2KeyPress(Sender: TObject; var Key: Char);
begin
if (Key = #13) then
begin
Label1.Caption := Edit1.Text;
Label2.Caption := Edit2.Text;
end;
end;

procedure TForm1.Button2Click(Sender: TObject);
var
TFile : TextFile;

begin
AssignFile(TFile, 'physics.txt');
Reset(TFile);
Append(TFile);
Writeln(TFile, Edit1.Text);
Writeln(TFile, Edit2.Text);
Writeln(TFile, ' ');
Flush (TFile);
CloseFile(TFile);

Button2.Caption := 'Done';
Edit1.ReadOnly := true;
Edit2.ReadOnly := true;
end;

procedure TForm1.Button5Click(Sender: TObject);
begin
Close;
end;

procedure TForm1.About1Click(Sender: TObject);
begin
Form2.ShowModal;
end;

procedure TForm1.Button3Click(Sender: TObject);
begin
Form3.Show;
end;

procedure TForm1.Button4Click(Sender: TObject);
begin
Form4.Show;
end;

end.



unit Unit3;

interface

uses
Windows, Messages, SysUtils, Classes, Graphics, Controls, Forms, Dialogs,
StdCtrls, ExtCtrls;

type
TForm3 = class(TForm)
Panel1: TPanel;
Button1: TButton;
Button2: TButton;
Button3: TButton;
Panel2: TPanel;
Image1: TImage;
Image2: TImage;
Image3: TImage;
Button4: TButton;
Button5: TButton;
Button6: TButton;
Button7: TButton;
Label1: TLabel;
Label2: TLabel;
Label3: TLabel;
Label4: TLabel;
Label5: TLabel;
Label6: TLabel;
Shape1: TShape;
Shape2: TShape;
Shape3: TShape;
procedure Button3Click(Sender: TObject);
procedure Button1Click(Sender: TObject);
procedure Button2Click(Sender: TObject);
procedure Button5Click(Sender: TObject);
procedure Button6Click(Sender: TObject);
procedure Button7Click(Sender: TObject);
private
{ Private declarations }
public
{ Public declarations }
function ActivateCard(pThisCard : integer) : boolean;
end;

type
TPlrCard = class
public
Name : string;
Expression : string;

constructor Create;
end;

type
TDisplay = class
public
Activated : boolean;

constructor Create;
end;

var
Form3: TForm3;
PlrCard : array[1..55] of TPlrCard;
Display : array[1..4] of TDisplay;

implementation

uses uMain, Unit4;

{$R *.DFM}

constructor TPlrCard.Create;
begin
end;

constructor TDisplay.Create;
var
i : integer;

begin
{set Display cards to false}
for i := 1 to 3 do
begin
Display[i].Activated := false;
end;

{read in values for the Plrcards}

{SetUpPlrCards;}
end;

procedure TForm3.Button3Click(Sender: TObject);
begin
Form1.Close;
end;

procedure TForm3.Button1Click(Sender: TObject);
begin
Form1.Show;
end;

procedure TForm3.Button2Click(Sender: TObject);
begin
Form4.Show;
end;


procedure TForm3.Button5Click(Sender: TObject);
begin
Display[1].Activated := ActivateCard(1);
end;


procedure TForm3.Button6Click(Sender: TObject);
begin
Display[2].Activated := ActivateCard(2);
end;

procedure TForm3.Button7Click(Sender: TObject);
begin
Display[3].Activated := ActivateCard(3);
end;

function TForm3.ActivateCard(pThisCard : integer) : boolean;
var
i : integer;

begin
if Display[pThisCard].Activated = true then
begin
if (pThisCard <> 4 ) then
Display[pThisCard].Activated := false;
case pThisCard of
1: Shape1.Brush.Color := clTeal;
2: Shape2.Brush.Color := clTeal;
3: Shape3.Brush.Color := clTeal;
4: begin
for i := 1 to 3 do
begin
Display[i].Activated := false;
end;
Shape1.Brush.Color := clTeal;
Shape2.Brush.Color := clTeal;
Shape3.Brush.Color := clTeal;
end;
end;
end
else
begin
Display[pThisCard].Activated := true;
case pThisCard of
1: Shape1.Brush.Color := clYellow;
2: Shape2.Brush.Color := clYellow;
3: Shape3.Brush.Color := clYellow;
4: begin
for i := 1 to 3 do
begin
Display[i].Activated := true;
end;
Shape1.Brush.Color := clYellow;
Shape2.Brush.Color := clYellow;
Shape3.Brush.Color := clYellow;
end;
end;
end;
Result := Display[pThisCard].Activated;
end;

end.

Lastly, your CD is in the mail. I have mailed it on the 12th of October. However, the mailing fees were pretty hefty this time at $72.00. This has caused me to cease future CD mailing until the project is done (even if you do volunteer to pay for the postage and handling fees yourself) Therefore, all future updates will be E-mailed to your E-Mail address or through Kasamba. The final product will be delivered on CD, when we have agreed that the work is complete. The price will be included in the already agreed upon sum of $350.00

My best wishes for our Mutual Success,
Amanda Dearheart

FOREWORD WALTER SPUNDE MATHS DEPT USQ.

At long last I found the time to dig through a portion of the mess in my tiny room here and find your typescript.It is ( obviously) enclosed herewith.
A harvard professor once chided a post graduate student who was particularly protective of his new found results.
"If you have got anything really original,"he told him "don't worry about hiding it: you'' have the devils own job to get anyone even just to listen".People aren't interested, and hence don't have time for original ideas.They only have time for things that will advance their careers,make them money or make them more comfortable.Original ideas seldom do any of those things.Books and ppers that extend well known ideas build the reputions and careers of others associated with those ideas.New ideas help no one.So it is a hard road to which you have set your foot.

To have an original idea is one thing,to have an original approach is even worse,but to get an idea across to people who don't want to hear is something else again that requires time and infinite patience.To try to get a new idea across using a new approach is to bang ones head against a brick wall.
Best wishes and good luck.
Walter Spunde.

Tuesday, September 25, 2007

THE LAW OF NON CONTRADICTION AND MATHS

This is the text version of the file http://www.math.psu.edu/simpson/papers/philmath.ps.
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Page 1
Logic and Mathematics
Stephen G. Simpson
Department of Mathematics
Pennsylvania State University
www.math.psu.edu/simpson/
April 30, 1999
This article is an overview of logic and the philosophy of mathematics. It is
intended for the general reader. It has appeared in the volume The Examined
Life: A Treasury of Western Philosophy, edited by Stanley Rosen and published
by the Book-of-the-Month Club.
Contents
1 Logic
2
1.1 Aristotelean logic . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.1.1 Subjects and predicates . . . . . . . . . . . . . . . . . . .
3
1.1.2 Syllogisms . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2 The predicate calculus . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2.1 Predicates and individuals . . . . . . . . . . . . . . . . . .
5
1.2.2 Formulas and logical operators . . . . . . . . . . . . . . .
6
1.2.3 Logical validity and logical consequence . . . . . . . . . .
7
1.2.4 The completeness theorem . . . . . . . . . . . . . . . . . .
9
1.2.5 Formal theories . . . . . . . . . . . . . . . . . . . . . . . .
9
2 Foundations of mathematics
10
2.1 The geometry of Euclid . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Formal theories for mathematics . . . . . . . . . . . . . . . . . . 12
2.2.1 A formal theory for geometry . . . . . . . . . . . . . . . . 12
2.2.2 A formal theory for arithmetic . . . . . . . . . . . . . . . 13
2.2.3 A formal theory of sets . . . . . . . . . . . . . . . . . . . 15
3 Philosophy of mathematics
17
3.1 Plato and Aristotle . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 The 20th century . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 The future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
References
20
1
1 Logic
Logic is the science of formal principles of reasoning or correct inference. Histor-
ically, logic originated with the ancient Greek philosopher Aristotle. Logic was
further developed and systematized by the Stoics and by the medieval scholastic
philosophers. In the late 19th and 20th centuries, logic saw explosive growth,
which has continued up to the present.
One may ask whether logic is part of philosophy or independent of it. Ac-
cording to Bochenski [2, §10B], this issue is nowhere explicitly raised in the
writings of Aristotle. However, Aristotle did go to great pains to formulate the
basic concepts of logic (terms, premises, syllogisms, etc.) in a neutral way, in-
dependent of any particular philosophical orientation. Thus Aristotle seems to
have viewed logic not as part of philosophy but rather as a tool or instrument
1
to be used by philosophers and scientists alike. This attitude about logic is
in agreement with the modern view, according to which the predicate calculus
(see 1.2 below) is a general method or framework not only for philosophical
reasoning but also for reasoning about any subject matter whatsoever.
Logic is the science of correct reasoning. What then is reasoning? According
to Aristotle [13, Topics, 100a25], reasoning is any argument in which certain
assumptions or premises are laid down and then something other than these
necessarily follows. Thus logic is the science of necessary inference. However,
when logic is applied to specific subject matter, it is important to note that
not all logical inference constitutes a scientifically valid demonstration. This is
because a piece of formally correct reasoning is not scientifically valid unless it
is based on a true and primary starting point. Furthermore, any decisions about
what is true and primary do not pertain to logic but rather to the specific subject
matter under consideration. In this way we limit the scope of logic, maintaining
a sharp distinction between logic and the other sciences. All reasoning, both
scientific and non-scientific, must take place within the logical framework, but
it is only a framework, nothing more. This is what is meant by saying that logic
is a formal science.
For example, consider the following inference:
Some real estate will increase in value.
Anything that will increase in value is a good investment.
Therefore, some real estate is a good investment.
This inference is logically correct, because the conclusion “some real estate is
a good investment” necessarily follows once we accept the premises “some real
estate will increase in value” and “anything that will increase in value is a
good investment”. Yet this same inference may not be a demonstration of its
conclusion, because one or both of the premises may be faulty. Thus logic can
help us to clarify our reasoning, but it can only go so far. The real issue in this
particular inference is ultimately one of finance and economics, not logic.
1
The Greek word for instrument is organon. The collection of Aristotle’s logical writings
is known as the Organon.
2
We shall now briefly indicate the basics of Aristotelean logic.
1.1 Aristotelean logic
Aristotle’s collection of logical treatises is known as the Organon. Of these
treatises, the Prior Analytics contains the most systematic discussion of formal
logic. In addition to the Organon, the Metaphysics
2
also contains relevant
material. See Aristotle [13] and Ross [19].
1.1.1 Subjects and predicates
Aristotelean logic begins with the familiar grammatical distinction between sub-
ject and predicate. A subject is typically an individual entity, for instance a man
3
or a house or a city. It may also be a class of entities, for instance all men. A
predicate is a property or attribute or mode of existence which a given subject
may or may not possess. For example, an individual man (the subject) may or
may not be skillful (the predicate), and all men (the subject) may or may not
be brothers (the predicate).
The fundamental principles of predication are:
1. Identity. Everything is what it is and acts accordingly. In symbols:
A is A.
For example, an acorn will grow into an oak tree and nothing else.
2. Non-contradiction. It is impossible for a thing both to be and not to be. A
given predicate cannot both belong and not belong to a given subject in a
given respect at a given time. Contradictions do not exist. Symbolically:
A and non-A cannot both be the case.
For example, an honest man cannot also be a thief.
3. Either-or. Everything must either be or not be. A given predicate either
belongs or does not belong to a given subject in a given respect at a given
time. Symbolically:
Either A or non-A.
For example, a society must be either free or not free.
2
The Metaphysics is Aristotle’s treatise on the science of existence, i.e., being as such. It
includes a detailed analysis of the various ways in which a thing can be said to be.
3
We use man in the traditional sense, equivalent to “human being”. There is no intention
to exclude persons of the female gender.
3
These principles have exercised a powerful influence on subsequent thinkers.
For example, the 20th century intellectual Ayn Rand titled the three main
divisions of her best-selling philosophical novel Atlas Shrugged
4
[18] after the
three principles above, in tribute to Aristotle.
1.1.2 Syllogisms
According to Aristotelean logic, the basic unit of reasoning is the syllogism. For
example, the real estate inference which was presented above is a syllogism. It
is of the form
Some A is B.
All B is C.
Therefore, some A is C.
Here A denotes real estate, B denotes increase in value, and C denotes a good
investment. Just as in the case of this example, every syllogism consists of two
premises and one conclusion. Each of the premises and the conclusion is of one
of four types:
universal affirmative:
All A is B.
universal negative:
No A is B.
particular affirmative: Some A is B.
particular negative:
Some A is not B.
The letters A, B, C are known as terms. Every syllogism contains three terms.
The two premises always share a common term which does not appear in the
conclusion. This is known as the middle term. In our real estate example, the
middle term is B, i.e., that which increases in value.
In order to classify the various types of syllogisms, one must take account of
certain symmetries. In particular, “no A is B” and “no B is A” are equivalent,
as are “some A is B” and “some B is A”. Furthermore, the order of the two
premises in a syllogism does not matter. Allowing for these symmetries, we
can enumerate a total of 126 possible syllogistic forms. Of these 126, only 11
represent correct inferences. For example, the form
all A is B, all B is C, therefore all A is C
represents a correct inference, while
all A is B, all C is B, therefore some A is C
does not.
The classification of syllogisms leads to a rather complex theory. Medieval
thinkers perfected it and developed ingenious mnemonics to aid in distinguishing
the correct forms from the incorrect ones. This culminated in the famous pons
asinorum (“bridge of asses”), an intricate diagram which illustrates all of the
syllogistic forms by means of a contrast between the good and the pleasurable.
See Bochenski [2, §24H, §32F].
4
A survey conducted for the Book-of-the-Month Club and the Library of Congress in 1991
found that Atlas Shrugged is the most influential book in the United States of America, second
only to the Bible. See http://www.lcweb.loc.gov/loc/cfbook/bklists.html.
4
1.2 The predicate calculus
In 1879 the German philosopher Gottlob Frege published a remarkable treatise,
the Begriffsschrift (“concept script”) [22]. This brilliant monograph is the origin
of modern logical theory. However, Frege’s account was defective in several
respects, and notationally awkward to boot. Instead of Frege’s system, we shall
present a streamlined system known as first-order logic or the predicate calculus.
The predicate calculus dates from the 1910’s and 1920’s. It is basic for all
subsequent logical research. It is a very general system of logic which accurately
expresses a huge variety of assertions and modes of reasoning. We shall see that
it is much more flexible than the Aristotelean syllogistic.
1.2.1 Predicates and individuals
In the predicate calculus, the subject/predicate distinction is drawn somewhat
differently from the way it is drawn in Aristotelean logic. The main point here
is that, in the predicate calculus, a subject is always an individual entity, never
a class of entities. For example, an individual man can be treated as a subject,
but the class of all men must be treated as a predicate. Since a subject in
the predicate calculus is always an individual entity, it is usual to speak of
individuals rather than subjects. We shall follow this customary practice.
The predicate calculus makes heavy use of symbolic notation. Lower-case
letters a, b, c, . . . , x, y, z, . . . are used to denote individuals. Upper-case letters
M, N, P, Q, R, . . . are used to denote predicates. Simple assertions may be
formed by juxtaposing a predicate with an individual.
For example, if M is the predicate “to be a man” and a is the individual
“Socrates”, then Ma denotes the assertion “Socrates is a man”. The symbol a
is called an argument of M. The predicate M may be applied to any individual,
and that individual is then an argument of M. If b is the individual “New
York”, then Mb asserts, falsely, that New York is a man. In general, if x is any
individual whatsoever, then Mx is the assertion that x is a man. This assertion
may or may not be true, depending on what x is. The expression Mx is called
an atomic formula of the predicate calculus.
Some predicates require more than one argument. For example, if B is the
predicate “bigger than”, then Bxy denotes the assertion “x is bigger than y”.
Thus B requires two arguments, and Bxy is an atomic formula. If we try to
use B with only one argument, we obtain something like Bx, i.e., “x is bigger
than”. This is not an atomic formula or any other kind of assertion. It is only
a meaningless combination of symbols. In analogy with English grammar, we
could say that Bxy is like a grammatically correct sentence, while Bx is merely
a sentence fragment. Such fragments play no role in the predicate calculus.
Let us now go into more detail about the role of individuals in the predicate
calculus. We have already said that lower-case letters denote individuals. We
now divide the lower-case letters into two groups: a, b, c, . .. near the beginning
of the alphabet, and x, y, z, . . . near the end of the alphabet. We insist on an
important grammatical or logical distinction between these two groups. Letters
5
of the first group are known as individual constants or simply constants. As
in the above examples, we think of them as denoting specific individuals, such
as Socrates or New York. Letters of the second group are known as individ-
ual variables or simply variables. For example, x is a variable. We think of
x as denoting not a specific individual but rather an arbitrary or unspecified
individual.
5
1.2.2 Formulas and logical operators
We have already mentioned two kinds of symbols: lower-case letters for individ-
uals (constants and variables), and upper-case letters for predicates. In addition
to these, the predicate calculus employs seven special symbols known as logical
operators
6
:
&
The names and meanings of the logical operators are given by
symbol
name
usage
meaning
&
conjunction
. . . & . . .
“both . . . and . . . ”
disjunction
. . . ∨ . . .
“either . . . or . . . (or both)”
negation
∼ . . .
“it is not the case that . . . ”
implication
7
. . . ⊃ . . .
“if . . . then . . . ”
bi-implication
8
. . . ≡ . . .
“. . . if and only if . . . ”
universal quantifier
∀x . . .
“for all x , . . . ”
existential quantifier
∃x . . .
“there exists x such that .. . ”
Here x is any variable.
A formula is a meaningful expression built up from atomic formulas by
repeated application of the logical operators. In the above table, an ellipsis
mark . . . stands for a formula within a larger formula.
For example, suppose we have a predicate M meaning “is a man”, another
predicate T meaning “is a truck”, and another predicate D meaning “drives”.
Here M and T are predicates which require only one argument apiece. The
predicate D requires two arguments: the driver, and the vehicle being driven.
5
The idea of using letters such as x and y as variables is of great value. Historically,
the creators of the predicate calculus borrowed this idea from the mathematical discipline
known as algebra. Recall that algebra is a kind of generalized arithmetic. In algebra there
are constants, i.e., specific quantities such as 2, the square root of 10, etc., but there are
also variables such as x, y, etc. The key idea of algebra is that a variable x represents an
unspecified or unknown quantity. It always stands for some quantity, but it may stand for any
quantity. The use of variables makes algebra much more powerful than arithmetic. Variables
help us to express and solve equations such as 2x + 3y = 11 involving one or more unknown
quantities. Variables can also be used to express arithmetical laws such as x + y = y + x.
6
The first five logical operators ( & , ∨, ∼ , ⊃, ≡) are equivalent to so-called “Boolean logic
gates” of electrical engineering. Formulas built from them may be viewed as representations
of the binary switching circuits that control the operation of modern digital computers. See
Mendelson [14, 15].
7
This is the so-called “material implication”: Φ
1
⊃ Φ
2
is equivalent to ∼ (Φ
1
& ∼ Φ
2
).
8
This is called bi-implication because Φ
1
≡ Φ
2
is equivalent to (Φ
1
⊃ Φ
2
) & (Φ
2
⊃ Φ
1
).
6
Thus Mx, Ty, and Dxy are atomic formulas meaning “x is a man”, “y is
a truck”, and “x drives y”, respectively. A typical formula built from these
atomic formulas is
∀x (Mx ⊃ ∃y (Ty & Dxy))
which we can translate as “for all x, if x is a man then there exists y such that
y is a truck and x drives y”. In other words,
Every man drives at least one truck.
Similarly, the formula
∀y (Ty ⊃ ∃x (Mx & Dxy))
translates to
Every truck is driven by at least one man.
In writing formulas, we often use parentheses as punctuation marks to in-
dicate grouping and thereby remove ambiguity. If parentheses were not used,
one could construe the formula ∼ Ty & Dxy in two logically inequivalent ways:
as (∼ Ty) & Dxy (“y is not a truck, and x drives y”), or as ∼(Ty & Dxy) (“y
is not a truck that x drives”). The parentheses allow us to choose the meaning
that we intend.
The predicate calculus is very rich in expressive power. For example, the four
Aristotelean premise types discussed in 1.1.2 can easily be rendered as formulas
of the predicate calculus. Letting A and B be predicates which require one
argument apiece, we have
universal affirmative
all A is B
∀x (Ax ⊃ Bx)
universal negative
no A is B
∀x (Ax ⊃ ∼ Bx)
particular affirmative some A is B
∃x (Ax & Bx)
particular negative
some A is not B
∃x (Ax & ∼ Bx)
In the second line of this table, the universal negative “no A is B” could have
been rendered equivalently as ∼∃x (Ax & Bx), or as ∀x (Bx ⊃ ∼Ax).
The above table may tend to gloss over a subtle but philosophically signifi-
cant difference between Aristotelean logic and the predicate calculus. Namely,
where Aristotelean logic views A as a subject and B as a predicate, the predi-
cate calculus views both A and B as predicates. This is typical of the different
perspectives involved. Aristotelean logic emphasizes the universal essences of
subjects or entities, while the predicate calculus elevates predicates to a position
of supreme importance.
1.2.3 Logical validity and logical consequence
A formula of the predicate calculus is said to be logically valid if it is necessarily
always true, regardless of the specific predicates and individuals involved. For
example, the three fundamental principles of Aristotelean logic (see 1.1.1 above)
correspond to formulas as follows:
7
Identity:
∀x (Ax ≡ Ax).
Non-contradiction: ∼∃x (Ax & ∼ Ax).
Either-or:
∀x (Ax ∨ ∼Ax).
These formulas are logically valid, because they are “necessarily” or “automat-
ically” or “formally” true, no matter what predicate may be denoted by the
symbol A.
The predicate calculus concept of logical validity subsumes the Aristotelean
syllogism. Each syllogism corresponds to a logically valid implication
1
& Φ
2
) ⊃ Ψ
where Φ
1
and Φ
2
are formulas expressing the two premises and Ψ expresses the
conclusion. For example, the syllogism
some A is B, all B is C, therefore some A is C
has a predicate calculus rendition
((∃x (Ax & Bx)) & (∀x (Bx ⊃ Cx))) ⊃ (∃x (Ax & Cx))
and this formula is logically valid.
More generally, a formula Ψ is said to be a logical consequence of a set of
formulas Φ
1
, . . . , Φ
n
just in case
1
& · · · & Φ
n
) ⊃ Ψ
is logically valid. Here Φ
1
, . . . , Φ
n
are premises and Ψ is a conclusion. This
is similar to the Aristotelean syllogism, but it is of wider applicability, because
the premises and the conclusion can be more complex. As an example, the 19th
century logician Augustus DeMorgan noted
9
that the inference
all horses are animals,
therefore, the head of a horse is the head of an animal
is beyond the reach of Aristotelean logic. Yet this same inference may be para-
phrased as “if all horses are animals, then for all x, if x is the head of some
horse then x is the head of some animal”, and this corresponds to a logically
valid formula
(∀y (Hy ⊃ Ay)) ⊃ (∀x ((∃y (Rxy & Hy)) ⊃ (∃y (Rxy & Ay))))
of the predicate calculus. Here H, A, R denote “is a horse”, “is an animal”,
“is the head of”, respectively. Thus DeMorgan’s conclusion is indeed a logical
consequence of his premise.
9
See however Bochenski [2, §16E].
8
1.2.4 The completeness theorem
Formulas of the predicate calculus can be exceedingly complicated. How then
can we distinguish the formulas that are logically valid from the formulas that
are not logically valid? It turns out that there is an algorithm
10
for recognizing
logically valid formulas. We shall now sketch this algorithm.
In order to recognize that a formula Φ is logically valid, it suffices to construct
what is known as a proof tree for Φ, or equivalently a refutation tree for ∼Φ.
This is a tree which carries ∼Φ at the root. Each node of the tree carries a
formula. The growth of the tree is guided by the meaning of the logical operators
appearing in Φ. New nodes are added to the tree depending on what nodes have
already appeared. For example, if a node carrying ∼(Φ
1
& Φ
2
) has appeared,
we create two new nodes carrying ∼ Φ
1
and ∼Φ
2
respectively. The thought
behind these new nodes is that the only way for ∼(Φ
1
& Φ
2
) to be the case is if
at least one of ∼ Φ
1
or ∼ Φ
2
is the case. Similarly, if a node carrying ∼∀x Ψ has
already appeared, we create a new node carrying ∼Ψ , where Ψ is the result of
substituting a new constant a for the variable x throughout the formula Ψ. The
idea here is that the only way for the universal statement ∀x Ψ to be false is if Ψ
is false for some particular x. Since a is a new constant, Ψ is a formula which
may be considered as the most general false instance of Ψ. Corresponding to
each of the seven logical operators, there are prescribed procedures for adding
new nodes to the tree. We apply these procedures repeatedly until they cannot
be applied any more. If explicit contradictions
11
are discovered along each and
every branch of the tree, then we have a refutation tree for ∼ Φ. Thus ∼Φ is
seen to be logically impossible. In other words, Φ is logically valid.
The adequacy of proof trees for recognizing logically valid formulas is a major
insight of 20th century logic. It is a variant of the famous completeness theorem,
first proved in 1930 by the great logician Kurt Godel [5, 22].
On the other hand, the class of logically valid formulas is known to be
extremely complicated. Indeed, this class is undecidable: there is no algorithm
12
which accepts as input an arbitrary formula Φ and outputs “yes” if Φ is logically
valid and “no” if Φ is not logically valid. In this sense, the concept of logical
validity is too general and too intractable to be analyzed thoroughly. There will
never be a predicate calculus analog of the pons asinorum.
1.2.5 Formal theories
The predicate calculus is a very general and flexible framework for reasoning. By
choosing appropriate predicates, one can reason about any subject whatsoever.
These considerations lead to the notion of a formal theory.
10
The details of this algorithm are explained in modern logic textbooks. Variants of it have
been programmed to run on digital computers. They form the basis of a system of computer
logic. See Fitting [4].
11
An explicit contradiction is a pair of formulas of the form Ψ, ∼ Ψ.
12
The algorithms in question may be implemented as Turing machine programs. This
undecidability result is known as Church’s theorem. See Mendelson [15].
9
In order to specify a formal theory, one first chooses a small collection of
predicates which are regarded as basic for a given field of study. These predicates
are the primitives of the theory. They delimit the scope of the theory. Other
predicates must be defined in terms of the primitives. Using them, one writes
down certain formulas which are regarded as basic or self-evident within the
given field of study. These formulas are the axioms of the theory. It is crucial
to make all of our underlying assumptions explicit as axioms. Once this has been
done, a theorem is any formula which is a logical consequence of the axioms. A
formal theory is this structure of primitives, axioms, and theorems.
As a frivolous example, we could envision a theory of cars, trucks, and
drivers. We would begin with some primitives such as C (“is a car”), T (“is a
truck”), D (“drives”), M (“is a man”), etc. We could then write down certain
obvious or self-evident axioms such as ∀x (Mx ⊃ ∼ Cx) (“no man is a car”),
∀x ((∃y Dxy) ⊃ Mx) (“every driver is a man”), etc. Then, within the con-
straints imposed by the axioms, we could investigate the logical consequence
relationships among various non-obvious assertions, such as
∼∃x (Mx & ∃y (Dxy & Cy) & ∃z (Dxz & Tz))
(“nobody drives both a car and a truck”). Additional predicates V (“is a vehi-
cle”) and P (“is a driver”) can be defined in terms of the primitives. The defining
axioms for V and P would be ∀y (V y ≡ (Cy ∨ Ty)) and ∀x (Px ≡ ∃y Dxy), re-
spectively. In this fashion, we could attempt to codify all available knowledge
about vehicles and drivers.
More seriously, one could try to write down formal theories corresponding
to various scientific disciplines, such as mechanics or statistics or law. In this
way one could hope to analyze the logical structure of the respective disciplines.
The process of codifying a scientific discipline by means of primitives and
axioms in the predicate calculus is known as formalization. The key issue here
is the choice of primitives and axioms. They cannot be chosen arbitrarily. The
scientist who chooses them must exercise a certain aesthetic touch. They must
be small in number; they must be basic and self-evident; and they must account
for the largest possible number of other concepts and facts.
To date, this kind of formal theory-building has been convincingly carried
out in only a few cases. A survey is in Tarski [21]. The most notable successes
have been in mathematics.
2 Foundations of mathematics
Mathematics is the science of quantity. Traditionally there were two branches
of mathematics, arithmetic and geometry, dealing with two kinds of quantities:
numbers and shapes. Modern mathematics is richer and deals with a wider
variety of objects, but arithmetic and geometry are still of central importance.
Foundations of mathematics is the study of the most basic concepts and
logical structure of mathematics, with an eye to the unity of human knowledge.
10
Among the most basic mathematical concepts are: number, shape, set, function,
algorithm, mathematical axiom, mathematical definition, mathematical proof.
The reader may reasonably ask why mathematics appears at all in this vol-
ume. Isn’t mathematics too narrow a subject? Isn’t the philosophy of mathe-
matics of rather specialized interest, all the more so in comparison to the broad
humanistic issues of philosophy proper, issues such as the good, the true, and
the beautiful?
There are three reasons for discussing mathematics in a volume on general
philosophy:
1. Mathematics has always played a special role in scientific thought. The
abstract nature of mathematical objects presents philosophical challenges
that are unusual and unique.
2. Foundations of mathematics is a subject that has always exhibited an
unusually high level of technical sophistication. For this reason, many
thinkers have conjectured that foundations of mathematics can serve as a
model or pattern for foundations of other sciences.
3. The philosophy of mathematics has served as a highly articulated test-
bed where mathematicians and philosophers alike can explore how various
general philosophical doctrines play out
13
in a specific scientific context.
The purpose of this section is to indicate the role of logic in the foundations
of mathematics. We begin with a few remarks on the geometry of Euclid. We
then describe some modern formal theories for mathematics.
2.1 The geometry of Euclid
Above the gateway to Plato’s academy appeared a famous inscription:
Let no one who is ignorant of geometry enter here.
In this way Plato indicated his high opinion of geometry. According to Heath
[9, page 284], Plato regarded geometry as “the first essential in the training of
philosophers”, because of its abstract character. See also Plato [17, Republic,
527B].
In the Posterior Analytics [13], Aristotle laid down the basics of the scientific
method.
14
The essence of the method is to organize a field of knowledge logically
by means of primitive concepts, axioms, postulates, definitions, and theorems.
The majority of Aristotle’s examples of this method are drawn from arithmetic
and geometry [1, 7, 9].
13
For example, philosophical intrinsicism may play out as mathematical Platonism. Philo-
sophical subjectivism may play out as mathematical constructivism. Nominalism may play
out as formalism.
14
Our modern notion of a formal theory (see 1.2.5 above) is a variant of Aristotle’s concept
of scientific method.
11
The methodological ideas of Aristotle decisively influenced the structure and
organization of Euclid’s monumental treatise on geometry, the Elements [8]. Eu-
clid begins with 21 definitions, five postulates, and five common notions. After
that, the rest of the Elements are an elaborate deductive structure consisting
of hundreds of propositions. Each proposition is justified by its own demon-
stration. The demonstrations are in the form of chains of syllogisms. In each
syllogism, the premises are identified as coming from among the definitions,
postulates, common notions, and previously demonstrated propositions. For
example, in Book I of the Elements, the demonstration of Proposition 16 (“in
any triangle, if one of the sides be produced, the exterior angle is greater than ei-
ther of the interior and opposite angles”) is a chain of syllogisms with Postulate
2, Common Notion 5, and Propositions 3, 4 and 15 (“if two straight lines cut
one another, they make the vertical angles equal to one another”) occurring as
premises. It is true that the syllogisms of Euclid do not always conform strictly
to Aristotelean templates. However, the standards of rigor are very high, and
Aristotle’s influence is readily apparent.
The logic of Aristotle and the geometry of Euclid are universally recognized
as towering scientific achievements of ancient Greece.
2.2 Formal theories for mathematics
2.2.1 A formal theory for geometry
With the advent of calculus in the 17th and 18th centuries, mathematics de-
veloped very rapidly and with little attention to logical foundations. Euclid’s
geometry was still regarded as a model of logical rigor, a shining example of
what a well-organized scientific discipline ideally ought to look like. But the
prolific Enlightenment mathematicians such as Leonhard Euler showed almost
no interest in trying to place calculus on a similarly firm foundation. Only in the
last half of the 19th century did scientists begin to deal with this foundational
problem in earnest. The resulting crisis had far-reaching consequences. Even
Euclid’s geometry itself came under critical scrutiny. Geometers such as Moritz
Pasch discovered what they regarded as gaps or inaccuracies in the Elements.
Great mathematicians such as David Hilbert entered the fray.
An outcome of all this foundational activity was a thorough reworking of
geometry, this time as a collection of formal theories within the predicate calcu-
lus. Decisive insights were obtained by Alfred Tarski. We shall sketch Tarski’s
formal theory for Euclidean
15
plane geometry.
16
As his primitive predicates, Tarski takes P (“point”), B (“between”), D
(“distance”), I (“identity”). The atomic formulas Px, Bxyz, Dxyuv, and Ixy
mean “x is a point”, “y lies between x and z”, “the distance from x to y is equal
to the distance from u to v”, and “x is identical to y”, respectively. Geometrical
15
Here “Euclidean geometry” refers to the familiar geometry in which the angles of a triangle
sum to 180 degrees, as distinct from the “non-Euclidean” (i.e., hyperbolic) geometry developed
by Bolyai and Lobachevsky in the 19th century.
16
Tarski also showed how to handle non-Euclidean plane geometry, as well as Euclidean and
non-Euclidean geometries of higher dimension, in a similar fashion.
12
objects other than points, such as line segments, angles, triangles, circles, etc.,
are handled by means of the primitives. For example, the circle with center x
and radius uv consists of all points y such that Dxyuv holds.
In geometry, two points x and y are considered identical if the distance
between them is zero. Tarski expresses this by means of an axiom
∀x ∀y ∀z (Dxyzz ⊃ Ixy).
Another axiom
∀w ∀x ∀y ∀z ((Bwxy & Bwyz) ⊃ Bxyz)
expresses the fact that, given any four points, if the second is between the first
and the third, and if the third is between the first and the fourth, then the third
is between the second and the fourth. A noteworthy axiom is
∀x ∀y ∀z ∀u ∀v ((Dxuxv & Dyuyv & Dzuzv & ∼Iuv) ⊃ (Bxyz ∨ Bxzy ∨ Byxz))
which says: any three points x, y, z equidistant from two distinct points u, v must
be collinear. This axiom is typical of two-dimensional (i.e., plane) geometry and
does not apply to geometries of dimension greater than two.
Altogether Tarski presents twelve axioms, plus an additional collection of
axioms expressing the idea that a line is continuous. The full statement of
Tarski’s axioms for Euclidean plane geometry is given at [10, pages 19–20]. Let
T
g
be the formal theory based on Tarski’s axioms.
Remarkably, Tarski has demonstrated that T
g
is complete. This means that,
for any purely geometrical
17
statement Ψ, either Ψ or ∼Ψ is a theorem of T
g
.
Thus we see that the axioms of T
g
suffice to answer all yes/no questions of
Euclidean plane geometry. Combining this with the completeness theorem of
Godel, we find that T
g
is decidable: there is an algorithm
18
which accepts as
input an arbitrary statement of plane Euclidean geometry, and outputs “true”
if the statement is true, and “false” if it is false. This is a triumph of modern
foundational research.
2.2.2 A formal theory for arithmetic
By arithmetic we mean elementary school arithmetic, i.e., the study of the
positive whole numbers 1, 2, 3, .. . along with the familiar operations of addition
(+) and multiplication (×). This part of mathematics is obviously fundamental,
yet it turns out to be surprisingly complicated. Below we write down some of
the axioms which go into a formal theory of arithmetic.
19
Our primitive predicates for arithmetic are N (“number”), A (“addition”),
M (“multiplication”), I (“identity”). The atomic formulas Nx, Axyz, Mxyz,
17
This means that all occurrences of variables x within the formula Ψ are within subformulas
of the form ∀x (Px ⊃ . . .) or ∃x (Px & . . .). Thus we are restricting attention to the realm of
geometry and excluding everything else.
18
Such algorithms have been implemented as computer programs. They are useful in
robotics and other artificial intelligence applications.
19
Two recent studies of formal arithmetic are Hajek/Pudlak [6] and Simpson [20].
13
Ixy mean “x is a number”, “x+y = z”, “x×y = z”, “x = y”, respectively. Our
axioms will use the predicates N, A, M, I to assert that for any given numbers
x and y, the numbers x + y and x × y always exist and are unique. We shall
also have axioms expressing some well known arithmetical laws:
substitution laws:
if x = y and x is a number then y is a number, etc.
commutative laws: x + y = y + x and x × y = y × x.
associative laws:
(x + y) + z = x + (y + z) and (x × y) × z = x × (y × z).
distributive law:
x × (y + z) = (x × y) + (x × z).
comparison law:
x = y if and only if, for some z, x + z = y or x = y + z.
unit law:
x × 1 = x.
Our formal axioms for arithmetic are as follows.
substitution laws:
∀x Ixx
∀x ∀y (Ixy ≡ Iyx)
∀x ∀y ∀z ((Ixy & Iyz) ⊃ Ixz)
∀x ∀y (Ixy ⊃ (Nx ≡ Ny))
existence and uniqueness of x + y :
∀x ∀y ∀z ∀u ∀v ∀w ((Ixu & Iyv & Izw) ⊃ (Axyz ≡ Auvw))
∀x ∀y ∀z (Axyz ⊃ (Nx & Ny & Nz))
∀x ∀y ((Nx & Ny) ⊃ ∃w ∀z (Iwz ≡ Axyz))
existence and uniqueness of x × y :
∀x ∀y ∀z ∀u ∀v ∀w ((Ixu & Iyv & Izw) ⊃ (Mxyz ≡ Muvw))
∀x ∀y ∀z (Mxyz ⊃ (Nx & Ny & Nz))
∀x ∀y ((Nx & Ny) ⊃ ∃w ∀z (Iwz ≡ Mxyz))
commutative laws:
∀x ∀y ∃z (Axyz & Ayxz)
∀x ∀y ∃z (Mxyz & Myxz)
associative laws:
∀x ∀y ∀z ∃u ∃v ∃w (Axyu & Auzw & Ayzv & Axvw)
∀x ∀y ∀z ∃u ∃v ∃w (Mxyu & Muzw & Myzv & Mxvw)
distributive law:
∀x ∀y ∀z ∃t ∃u ∃v ∃w (Ayzt & Mxtw & Mxyu & Mxzv & Auvw)
comparison law:
∀x ∀y ((Nx & Ny) ⊃ (Ixy ≡ ∼ ∃z (Axzy ∨ Ayzx)))
unit law:
∃z (Nz & (∼∃x ∃y Axyz) & ∀w (Nw ⊃ Mwzw))
Let T
a
be the formal theory specified by the above primitives and axioms.
It is known that T
a
suffices to derive many familiar arithmetical facts. For
example, 2+2 = 4 may be expressed, awkwardly
20
to be sure, as (1+1)+(1+1) =
20
This kind of awkwardness can be alleviated by means of various devices. In particular,
14
((1 + 1) + 1) + 1 or
∃x ∃y ∃z ∃w (Mxxx & Axxy & Axyz & Axzw & Ayyw)
21
and this formula is indeed a theorem of T
a
, i.e., a logical consequence of the
axioms of T
a
. Another theorem of T
a
is
∀x ∀y ∀z ∀w (((Axzw & Ayzw) ∨ (Mxzw & Myzw)) ⊃ Ixy)
expressing a familiar cancellation law: if either x + z = y + z or x × z = y × z,
then x = y.
On the other hand, the axioms of T
a
are by no means exhaustive. They
can be supplemented with other axioms expressing the so-called mathematical
induction or least number principle: if there exists a number having some well-
defined property, then among all numbers having the property there is a smallest
one. The resulting formal theory is remarkably powerful, in the sense that its
theorems include virtually all known arithmetical facts. But it is not so powerful
as one might wish. Indeed, any formal theory which includes T
a
is necessarily
either inconsistent
22
or incomplete. Thus there is no hope of writing down
enough axioms or developing an algorithm to decide all arithmetical facts. This
is a variant of the famous 1931 incompleteness theorem of Godel [5, 22]. There
are several methods of coping with the incompleteness phenomenon, and this
constitutes a currently active area of research in foundations of mathematics.
The contrast between the completeness of formal geometry and the incom-
pleteness of formal arithmetic is striking. Both sides of this dichotomy are of
evident philosophical interest.
2.2.3 A formal theory of sets
One of the aims of modern logical research is to devise a single formal theory
which will unify all of mathematics. Such a theory will necessarily be subject
to the Godel incompleteness phenomenon, because it will incorporate not only
T
g
but also T
a
.
One approach to a unified mathematics is to straightforwardly embed arith-
metic into geometry, by identifying whole numbers with evenly spaced points
on a line. This idea was familiar to the ancient Greeks. Another approach is
to explain geometry in terms of arithmetic and algebra, by means of coordinate
systems, like latitude and longitude on a map. This idea goes back to the 17th
century mathematician and philosopher Rene Descartes and the 19th century
mathematician Karl Weierstrass. Both approaches give rise to essentially the
same formal theory, known as second-order arithmetic.
23
This theory includes
both T
a
and T
g
and is adequate for the bulk of modern mathematics. Thus the
standard mathematical notation such as x+2 can be incorporated into the predicate calculus.
21
In this formula, the variables x, y, z, w play the role of 1, 2, 3, 4 respectively.
22
A formal theory is said to be inconsistent if its axioms logically imply an explicit contra-
diction. Such theories are of no scientific value.
23
A recent study of second-order arithmetic is Simpson [20].
15
decision about whether to make geometry more fundamental than arithmetic or
vice versa seems to be mostly a matter of taste.
A very different approach to a unified mathematics is via set theory. This
is a peculiarly 20th century approach. It is based on one very simple-looking
concept: sets. Remarkably, this one concept leads directly to a vast structure
which encompasses all of modern mathematics.
A set is a collection of objects called the elements of the set. We sometimes
use informal notations such as x = {y, z, . . .} to indicate that x is a set consisting
of elements y, z, . .. . The number of elements in a set can be arbitrarily large
or even infinite. A basic principle of set theory is that a set is determined
by its elements. Thus two sets are identical if and only if they have the same
elements. This principle is known as extensionality. For example, the set {a, b, c}
is considered to be the same set as {c, a, b} because the elements are the same,
even though written in a different order.
Much of the complexity of set theory arises from the fact that sets may
be elements of other sets. For instance, the set {a, b} is an element of the set
{{a, b}, c} and this is distinct from the set {a, b, c}.
For a formal theory of sets, we use three primitives: S (“set”), I (“identity”),
E (“element”). The atomic formulas Sx, Ixy, Exy mean “x is a set”, “x is
identical to y”, “x is an element of y”, respectively. One of the ground rules of
set theory is that only sets may have elements. This is expressed as an axiom
∀w ∀x (Ewx ⊃ Sx). In addition there is an axiom of extensionality
∀x ∀y ((Sx & Sy) ⊃ (Ixy ≡ ∀w (Ewx ≡ Ewy)))
and an axiom ∃x (Sx & ∼∃w Ewx) expressing the existence of the empty set,
i.e. a set { } having no elements. A list of all the axioms of set theory may be
found in textbooks [11, 15]. Let T
s
be the formal theory of sets based on these
axioms.
The set theory approach to arithmetic is in terms of the non-negative whole
numbers 0, 1, 2, 3, . . .. These numbers are identified with specific sets. Namely,
we identify 0 with the empty set { }, 1 with {{ }}, 2 with {{ }, {{ }}}, 3 with
{{ }, {{ }}, {{ }, {{}}}}, etc. In general, we identify the number n with the set
of smaller numbers {0, 1, . . ., n − 1}. Among the axioms of T
s
is an axiom of
infinity asserting the existence of the infinite set ω = {0, 1, 2, 3, . . .}. One can
use the set ω to show that T
s
includes a theory equivalent to T
a
. After that, one
can follow the ideas of Descartes and Weierstrass to see that T
s
also includes a
theory equivalent to T
g
. It turns out that the rest of modern mathematics can
also be emulated within T
s
. This includes an elaborate theory of infinite sets
which are much larger than ω.
The set-theoretical approach to arithmetic and geometry is admittedly some-
what artificial. However, the idea of basing all of mathematics on one simple
concept, sets, has exerted a powerful attraction.
24
The implications of this idea
are not yet fully understood and are a topic of current research.
24
The idea of set-theoretical foundations gave rise to the “new math” pedagogy of the
1960’s. For a lively discussion, see Kline [12].
16
3 Philosophy of mathematics
In this section we indicate some issues and trends in the philosophy of mathe-
matics.
3.1 Plato and Aristotle
The objects that are studied in mathematics tend to be somewhat abstract
and remote from everyday perceptual experience. Therefore, the existence and
nature of mathematical objects present special philosophical challenges. For
example, is a geometrical square different from a square floor tile? If so, then
where is the geometrical square? Is it on the floor, in our minds, or somewhere
else? And what about sets? Is a set of 52 cards something other than the cards
themselves?
The ancient Greek philosophers took such questions very seriously. Indeed,
many of their general philosophical discussions were carried on with extensive
reference to geometry and arithmetic. Plato seemed to insist that mathematical
objects, like the Platonic forms or essences, must be perfectly abstract and have
a separate, non-material kind of existence. Aristotle [1, 7, 13, 19] dissected and
refuted this view in books M and N of the Metaphysics. According to Aristotle,
the geometrical square is a significant aspect of the square floor tile, but it can
only be understood by discarding other irrelevant aspects such as the exact
measurements, the tiling material, etc. Clearly these questions provide much
food for philosophical analysis and debate.
3.2 The 20th century
In the 20th century, the advent of the predicate calculus and the digital com-
puter profoundly affected our view of mathematics. The discovery that all of
mathematics can be codified in formal theories created a huge stir. One expres-
sion of this excitement was the rise of an extreme philosophical doctrine known
as formalism.
25
According to formalism, mathematics is only a formal game, concerned solely
with algorithmic manipulation of symbols. Under this view, the symbols of the
predicate calculus do not denote predicates or anything else. They are merely
marks on paper, or bits and bytes in the memory of a computer. Therefore,
mathematics cannot claim to be any sort of knowledge of mathematical objects.
Indeed, mathematical objects do not exist at all, and the profound questions
debated by Plato and Aristotle become moot. Mathematics is nothing but a
kind of blind calculation.
The formalist doctrine fits well with certain modern trends in computer
science, e.g., artificial intelligence. However, formalism has proved inadequate as
an integrated philosophy of mathematics, because it fails to account for human
mathematical understanding, not to mention the spectacular applications of
mathematics in fields such as physics and engineering.
25
See for example Curry [3].
17
By way of reaction against formalism, several alternative doctrines have been
advocated. One of these is constructivism, the idea that mathematical knowl-
edge can be obtained by means of a series of purely mental constructions. Under
this view, mathematical objects exist solely in the mind of the mathematician,
so mathematical knowledge is absolutely certain. However, the status of math-
ematics vis a vis the external world becomes doubtful. An extreme version of
constructivism is so solipsistic that it does not even allow for the possibility of
mathematical communication from one mind to another.
An additional disturbing feature of constructivism is that it entails rejection
of the basic laws of logic. To see how this comes about, consider some specific
mathematical problem or question
26
of a yes/no nature, for which the answer is
currently unknown. (Mathematics abounds with such questions, and the Godel
incompleteness phenomenon suggests that such questions will always exist.)
Express the “yes” answer as a formula Ψ and the “no” answer as the negated
formula ∼Ψ. Since the answer is unknown, neither Ψ nor ∼ Ψ is in the mind of
the mathematician. Therefore, according to the constructivists, the disjunction
Ψ∨∼Ψ is not a legitimate mathematical assumption. Thus Aristotle’s either-or
principle (see 1.1.1 and 1.2.3 above) must be abandoned.
27
Constructivism has the merit of allowing human beings to possess mathe-
matical knowledge. However, the constructivist rejection of the external world
and of Aristotelean logic are highly unpalatable to most mathematicians and
mathematically oriented scientists. For this reason, constructivism remains a
fringe movement on the 20th century mathematical landscape.
Another 20th century philosophical doctrine has arisen from set-theoretical
foundations. The reliance on infinite sets suggests many perplexing questions.
What do such sets correspond to in reality? Where are they, and how can the
human mind grasp them? In order to boldly answer these questions, and as a
reaction against formalism, many researchers in axiomatic set theory have sub-
scribed to what is known as set-theoretical Platonism. According to this variant
of the Platonic doctrine, infinite sets exist in a non-material, purely mathemat-
ical realm. By extending our intuitive understanding of this realm, we will be
able to cope with chaos issuing from the Godel incompleteness phenomenon.
The most prominent and frequently cited authority for this kind of Platonism
is Godel himself [5].
There is a good fit between set-theoretical Platonism and certain aspects of
20th century mathematical practice. However, as a philosophical doctrine, set-
theoretical Platonism leaves much to be desired. Many of Aristotle’s objections
to the Platonic forms are still cogent. There are serious questions about how a
theory of infinite sets can be applicable to a finite world.
We have mentioned three competing 20th century doctrines: formalism, con-
structivism, set-theoretical Platonism. None of these doctrines are philosoph-
ically satisfactory, and they do not provide much guidance for mathematically
oriented scientists and other users of mathematics. As a result, late 20th century
26
For example, we could consider the following difficult question of Goldbach. Can every
even number greater than 2 be expressed as the sum of two prime numbers?
27
See the essays of Brouwer and Kolmogorov in [22].
18
mathematicians have developed a split view, a kind of Kantian schizophrenia,
which is usually described as “Platonism on weekdays, formalism on weekends”.
In other words, they accept the existence of infinite sets as a working hypothesis
in their mathematical research, but when it comes to philosophical speculation,
they retreat to a formalist stance. Thus they have given up hope of an integrated
view which accounts for both mathematical knowledge and the applicability of
mathematics to physical reality. In this respect, the philosophy of mathematics
is in a sorry state.
3.3 The future
From the Renaissance through the 20th century, Aristotle’s ideas about the
nature of mathematical objects have been neglected and ignored. Now the
time seems ripe for a renovation of the philosophy of mathematics, based on
Aristotelean and neo-Aristotelean [16] ideas and bolstered by the techniques of
modern logic, including the predicate calculus.
The great mathematician David Hilbert anticipated such a renovation in his
1925 essay, On the Infinite [22]. Hilbert was aware that, according to modern
physics, the physical universe is finite. Yet infinite sets were playing an increas-
ingly large role in the mathematics of the day. Hilbert therefore recognized that
the most vulnerable chink in the armor of mathematics was the infinite. In order
to defend what he called “the honor of human understanding”, Hilbert proposed
to develop a new foundation of mathematics, in which formal theories of infinite
sets, such as T
s
, would be rigorously justified by reference to the finite. This is
Hilbert’s program of finitistic reductionism.
28
Although Hilbert did not cite Aristotle, we can imagine that Hilbert would
have profited from an examination of Aristotle’s distinction between actual and
potential infinity. An actual infinity is something like an infinite set regarded
as a completed totality. A potential infinity is more like a finite but indefinitely
long, unending series of events. According to Aristotle, actual infinities cannot
exist, but potential infinities exist in nature and are manifested to us in various
ways, for instance the indefinite cycle of the seasons, or the indefinite divisibility
of a piece of gold.
In any case, it turned out that Hilbert had stated his program in too sweeping
a fashion. The wholesale finitistic reduction which Hilbert desired cannot be
carried out. This follows from Godel’s incompleteness theorem [5, 22]. The
remarkable results obtained by Godel in 1931 caused the philosophical ideas of
Hilbert’s 1925 essay to fall into disrepute. Hilbert’s grand foundational program
appeared to be dead, broken beyond hope of repair.
The last 20 years have seen a revival of Hilbert’s program. Recent founda-
tional research [20] has revealed that, although T
s
is not finitistically reducible,
there are other formal theories which are finitistically reducible, in the precise
28
Hilbert is often inaccurately described as a formalist. The details of Hilbert’s program
will not be presented here, but see [20, 22]. Roughly speaking, a formal theory is said to be
finitistically reducible if it can be embedded into some very restricted formal theory such as
T
a
, which is physically meaningful and makes absolutely no reference to actual infinity.
19
sense envisioned by Hilbert. Moreover, these other formal theories turn out to
be adequate for a very large portion of mathematics. They do not encompass
actual infinities such as ω, but they do include the main results of arithmetic
and geometry and allied disciplines.
This new research has not yet had an impact on the philosophy of mathe-
matics or on mathematical practice. Philosophers and mathematicians are free
to choose which directions to pursue and which techniques to emphasize. Only
time will reveal the future evolution of the philosophy of mathematics.
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21