0.
Two Given Styles Of Lectures

1.
The Ampere/Voltage/Ohm

2.
The Gödel Numbers

3.
The Turing Machine

4.
The Nash Equilibrium

5.
More than Analogy

6.
The Greek Example

*0. Two Given Styles Of Lectures*

There
are two lectures, which are confidently designed for the listener.
One is extremely short and to the point, and actually has some hope
of being understood in real time. These are “do” lectures. The
other one is extremely long and will have to be absorbed over time,
and contains hints of things yet to come. These are “think”
lectures; both of these are badly done because training a lecture to
do this is not an immediate process.

And
who knows, you might actually learn something, even if the lecture is
disabled.

Obviously
this is the second kind of lecture, and the lecturer does not think
that all you who will get it will not get right way. In fact I would
be surprised if any of them got it, though one or two of you just
might. Given that this is a long lecture, and does not come with an
immediacy of understanding, it means that you will take time to
really understand. So let us begin, first with the obvious parallel
to a battery, then by going through the three talking points, then
closing with a distinct problem – and some questions.

As
I said, you are not going to get this at once. But that is all right,
because now that you know the information: you can digest at your own
pace, and not rush to produce it on a test. You are at the point of
having to digest at your own pace, and not by the cycles of semester
or quarterly information. While not everyone will treat you as
grown-up, I will, and it takes you days or weeks or months to get
this, that is all right by me. After all it took me a great deal of
time to come up with this, so it is going to take you time to
understand it.

First
of all, though, I would like to thank everyone involved in giving
these lectures, because it is tough to get together a lecture that
will be understood. And each and every one involved deserves a hand.
And I would also like to thank all of you for attending these
lectures, because they are not easy to absorb.

*1. The AVO*

By
now we know what AVO is: an Ampere driven by a Volt, against a
resistance of one Ohm. And if we do not there are a large number of
websites willing to explain this for you. We do not need to know
everything about physics, just where to find that stuff that we do
not know when to know it. The AVO is the basic unit of electricity,
and everything about electricity can be expressed by it, or by
numbers which come the same thing, usually with a square somewhere in
the calculation. Is so basic and so obvious that it will take four
years to just know what you are looking for, and another five or so
to actually make a contribution to the field of EE.

What
this means, if you think about it, this field is rich in density with
possibility, and produces interactions that are complex and
extraordinary. You can spend your entire life working with just such
equations and workings out of the field. Airplanes, automobiles,
trains, and all manner of appliances – from very large to the very
small screen out for just a little nudge of EE. This means that a
triangle of forces is at work in any complete electrical circuit.
Even if Ohms is zero, because after all zero is a number.

When
first you start experimenting with electricity, everything seems
novel – and if you will excuse the expression, shocking. That is
because electricity goes where it wants to, rather than where you
wanted it to go. And that means you have to learn the language of
electricity if you want it to go anywhere. But there is a saving
grace – and that is, it will ignore, most of the time, detours
which do not interest it. Compare this with water, which will squeeze
out any little gap and come flushing out of every pore if you give it
a chance – and even if you do not think there is anything to
squeeze. Water will find a way – whereas electricity will not.

What
makes electricity so useful is that it provides a framework for
introducing a novel concept: GTN as a unit of measurement. Because if
you think about it, it is basically the same concept: you have a way
of current running on a system which will work if the current times
the system is over the backlog. But what has not been made clear to
everyone is what GTN actually works out to being.

So
let us take a moment to review the AVO, and then introduce the
information equivalent. This way if you get lost in the world of GTN,
you can slip back to an old standby and reason by analogy. So what is
AVO?

Let
us start with amperage, and what it is actually doing. Amperage is
the amount of electrical current present at a given point, When it is
expressed the current is in Amps, whereas the charge accumulated is
measured in coulombs. But that basically means that a charge will not
move anywhere unless there is voltage which is expressed as kilogram
meters squared over Amperes times seconds cubed. In other words a
weight times a distance over seconds cube. This is more easily
expressed as Watts over Amperes. Now here is where physicists divide
from electronic engineers. Physicist will know that there is some
conversion factor which he will remember two digits of, or look up if
more precise is needed – where as the electronic engineer will know
the precise nine digit, and have them memorized. And if you look at
their phone, it will be (xxx) 483-5979. They will even take issue
with the water flow analogy, because everyone in the EE space knows
that water flow is not the same thing.

Then
everything slips in two places, the average does the work, the
voltaic is the hurdle to cross, and

**Ohm is the resistance between two points on a conductor. Thus Voltage, Amper, and Ohmage are nice neat one-for-one equations.**
But
there is a small catch: it took a little over 100 years to do this.
Which seems a bit long for some rather simple equations, not one of
them uses more than cubed. Which brings me to what is new here, not
defining one cell by a current, but by information.

So
for the moment hold on to Ampere, Volt, and Ohm in a triangle which
is the position in current space.

*2. In the Beginning, There was Gödel*

First
it is unclear why one would even need to do this, but Cantor showed
us that it was necessary at least in terms of mathematics. It did not
occur to anyone that this would have practical possibilities, at
least so far as the word willing to publish, or even quoted. Now if I
were relating all of the history of this idea I would begin with
Cantor, move on to bring Principia Mathematic by the pair of
misguided geniuses Whitehead and Russell, then take a moment to
admire the foundation of the rules of the game by von Neumann and
Morgenstern, and tell all of those odd quirks of personality. This is
what one gets by thinking about probability vectors such as P and Q
and a positive number that will solve an equation that is
complementary to p

^{T }(A- lB)q = 0.
But
since a great many of people are doing this, and much better than I
can, I will restrain myself to the three main characters of my drama.
The first of these is a man who wandered on to the scene at 23, and
proved that what was impossible was in fact required: Gödel.

In
1931, he began with the assertion that “the development of
mathematics has been in the direction of greater exactness has – as
is well known – lead to large tracts of it becoming well
formalized, so that proves again be carried out according to a new
mechanical rules, the most comprehensive formal systems yet set up
are, on the one hand, the system of PM, and on the other hand, the
system for set theory by Zermelo-Fraekel (later extended by von
Neumann).” I will note that this is not the last time that von
Neumann is referenced. I will also mention the 1997 extract by
Schwalbe and Walker as a good summary of Zermelo role in all of this,
by Harvard University, with input from numerous inputs, including
Kuhn.

^{1}^{}

Rather
than delaying until he has a proof that he is going to show is false,
he asks directly the question which others had taken on in large
volumes - and replies that it is not the case. He then says that it
is obvious, which is like a beer tromping through the woods with a
picnic basket in its mouth, everyone will wonder who made it, because
it was not the bear.

He
then sets off a proof with one free variable, which can be any
natural number. And denotes Bew X as being a provable formula for X.
from here he directly attacks the provability of Bew. While the proof
is by no means the easiest one to prove, each of the ones that come
after it know that it is already too be proved and simply are trying
to form a better solution to something already proven. The thing
about Gödel proof is that he was not sure that he was going to do
this.

What
now follows is quite tricky, he sets up a series of proofs that means
that a, insert cases, is neither confirmable nor deniable within the
system, which is, as is noted, equivalent to a PM system. In other
words certain a are not provable nor disprovable in any finite number
of steps. One can do this proof more simply, and I allow you to try
your hand at it, because I did, as did several others. But there is
something about this original proof, which stands up to the test of
time.

One
thing about it is that it is on the backside of one to one proof.
There may be proofs that are shorter, but there is no proof, which is
easier to explain then the original. This is partially because every
proof is recursive, and that means that you simply have to go down
the proof, until you hit the bottom. In other words it not only
proves its point, but also proves the next two pieces have to be
there as well. That is to say: Turing and Nash. And this is all in a
very short book. You would be well advised to purchase “On Formally
Undecidable Propositions of Principia Mathematica and Related
Systems”

^{2}, and run through it yourselves, because as Newton's three principles of motion, and a universal gravity, were in its day, this is one of the seminal moments of ours.
This
proof then turns on the uniqueness of prime numbers; it translates
into primes, and builds from there into a GN version, which will be
unique. It then says for any proof system one of three things is
possible:

1.
The system is finite, and will cast out all of the proofs which would
create disharmonies.

2.
The system is finite or infinite with a fixed number of axioms, but
will have disharmonies within it.

3.
The system is infinite in its number of axioms.

One
might think, but not to quickly mind you, that a system which is
finite and casts out all of the proofs would be preferable. Humanity
got along just fine without them, or so it appears. After all
classical logic has only two variables, and a virtual plot of
eliminating the obvious from the conclusion. “Whatever is left,
however improbable, is the truth,” says Sherlock. But the things
that you have to give up are tremendous. Such as multiplication, that
is right 3 x 3 is right out. Sherlock may get by without
multiplication, but the rest of us need it, for buying milk, and so
on.

The
third choice offers attractions which have yet to be explored, but
will require things like Goldbach Postulate will have to be included,
things on which nothing else relies upon. This is quite a drag on the
system, and it will have to have a new sort of mathematics to engage
in, there might not be anything wrong with that, and one of you might
discover it.

For
this reason, the second, is why Codd and Date needed variables which
were undecided in their Relational Database System, a direct
descendent of what we are talking about.

One
major difference between a Gödel numbered system and a current is
that generally it is the GN,
which is to be found, either because it is unknown, or it is being
hidden by the party who wants not to reveal what they are doing. This
means that GN is often worked backwards from a von Neumann position,
the proof of which is the problem to be set up.

One
example of this is from my friend Scott Kominers

^{3}, of the Harvard Society of Fellows. The key determiner of a patent troll, called in the language of art a “Non-Practicing Entity”, is the difference in the target company. In a normal lawsuit there is harm that is alleged to have been done, and a lawsuit is filed against the holder of the patent. In a NPE lawsuit however there is a direct correlation between the ability to pay, and the likelihood that such a lawsuit will be filed. Even if the money does not come from the patent.
This,
and other examples, means that the GN is only a fraction of the
story, and for all practical purposes is the endpoint not the
beginning. One often finds this in the sciences: first one needs to
find out where there is a problem. But it also means that the problem
exists and no way of solving it has yet appeared. It is a problem
that is discovered first.

But
when a problem appears, most people do not think of how to solve the
problem in general, but attack the problem in detail; which clutters
the literature with innumerable small examples. But a few
pioneers see a larger question, and set themselves to work. Once a
current has been made, in our electoral example, then one needs a
voltage and resistance. In the same way, in informational space, one
needs a machine, and a limit. These two followed quickly because once
the problem was determined to be informational, it was only a matter
of time and brilliance. Because let us remember that even the
failures were brilliant, and even more so the two successes.

*3. The Turing Machine*

But
on the ground, there is a problem, which is looming in the thoughts
of the participants: they see problems everywhere, and there seems to
be nothing to relate them. It is a fog that no one can see their way
out of. A large part of the problem is that no one realizes that the
mechanics of a new generation are fundamentally different from those
in the past. This was the same with light and electricity, because no
one knew what the difference was. And once it was found it took 50
years to discover the means by which it was enforced by nature. Now
of course the Higgs will be known to all, but it was not that way for
decades.

In
the information space, the idea that it was a machine was not even in
people's mind. Except one, who had been working on cleaning up the
mathematics of Gödel, and he discovered a seemingly bizarre
creation. Remember that no one had thought of it as a machine, even
though the concept had turned up in Ancient Greece, in Renaissance
France, and in Victorian England. But each time it had been
dismissed, and the machines slumbered quietly until someone picked up
the pieces.

^{4}^{}

The
man of course was Turing, who was at the right place with the right
brain power. Because there were people who saw the signs and ignored
them. As Winston Churchill said, they hit upon the truth and get up
on their way as if nothing happened.

But
Turing was working on the machines, and saw a formula, which would
work: an unlimited memory, a scanned symbol, and a simple set of
instructions. And that is all that needs to be accomplished. All of
the rest of the computer hardware that we possess is towards making
it run better. He invented this in 1936, a few years after the GN was
invented. Almost nobody else even thought of a machine for this
problem.

But
what this did is simply astonishing, partially because it solves the
riddle of the Enigma – whose German scientists thought it to be
unbreakable. Yet when the machine was perfected, that is a real
machine, it was able to break things on and enigma code machine
almost in real time. But it was a problem known to David Hilbert, and
put on his list in 1900, called, appropriately enough, the “Hilbert's
Problems”. Thus the uncrackable problem was indeed cracked by a
simple machine. Remember that the mathematics behind this question
was quite exact: first of all was the mathematics complete, second of
all was the mathematics consistent, and third of all was the
mathematics decidable?

Then
on a day in Grantchester Turing had what he needed, because he knew
that this problem was related to a universal machine. Once he had
that concept, it was just working out what the machine had to do. And
Turing had been the one man who was determined to see that a
typewriter was far more than a mechanical device to produce letters.

All
that was to be done is calculating the equivalent of Ohms. But again
remember that the people at the time did not see this at all, because
what they thought of as problems – rather than thinking of
solutions – gripped their minds tightly. So we now move on to Nash,
and one of the most brilliant papers of all time, and one of the most
surprising.

So
even as you play with Google, realize there is a long fight from 1900
when Hilbert enunciated the question, until you find what you are
looking for by typing in a view words into a box.

*4. The Nash Equilibrium*

Each
type of genius is unique to itself. Gödel took two seemingly
unrelated things, improved they were related to each other. He did
this even though two geniuses of field tried to do things the
opposite way – and Whitehead and Russell were geniuses. Turing was,
for all intensive purposes, a mystic – seeing things in a
completely different fashion, even though the objects that he reached
with were right there. They just thought of them as machines, whereas
he understood them to be computers. Finally, we reach Nash, whose
gift was to see things in place site. Remember that Gödel himself
see this though he was a resident at Princeton at the time, but did
recognize for what was. It can easily be said that genius has the
gift of a complex mind, but is trumped by a genius with a simple
mind. So let us get to Nash simple genius, which is often overlooked
next to the complex geniuses that surround him.

Though
I would like to mention that in 1951 he was hired by this
institution, to be a C. L. E. Moore instructor in the mathematics
faculty, and eventually a full professorship.

^{5}^{}

So
what exactly does he prove with his famous Nash Equilibrium? Is only
a few simple steps, and it proves two things: first that the group
that is a Nash equilibrium is concave rather than convex, and second
one does not prove it forward but backwards. Everything else in the
group is devoted to one of these two concepts. You might think that
anyone can prove this, and you would be right. Anyone can do this
once knows that it has been proved, but no one can prove this if he
does not know that the proof is sitting right there in front of them.
Only one person could figure that out, even Gödel did not figure
that out.

Now
we have we have to ask: why is this important?

Let
us examine the proof, looking for the key texts, which are important.

First
he finds an equilibrium point such that if and only if for every
individual high there is only one option, for every strategy that the
others choose on. That is, whatever your opponents choose, there is
only one choice for you, as opposed to one choice in that they choose
one way, and a different choice if they choose a different way. This
is powerful, because you do not have to concern yourself with what
they are deciding. It means that in best case for doing well, and in
worst case you are losing least.

Then
says that an equilibrium point it can be expressed as pairs of use
functions. By this he equates a single equilibrium point to its group
on a line. So an equilibrium point stands in for a set of values,
which is hard to grasp at first, but settles in your mind once have
grasped. Again I will remind you that this will be grasped only with
time. That a single point can stand in a curve is not to grasp.

Then
he makes a leap into two unknowns, based on Kakutani generalized
fixed point theorem. Though cleans it up by making reference to
Brouwer. Then he proceeds to show that all of the equilibrium points
are connected.

First
he shows that every finite game has an equilibrium point. This is
obvious, until you actually try and prove it. Then a web comes over
you. What Nash proves is that the top part of the equation is the s
part with pi as the numerator, over 1 plus the whole amount. Which is
hardly obvious.

What
he then must show is that the fixed points of the mapping are also
equilibrium points, which looks non-trivial, but has a solution. He
shows that under Brouwer, the cell must have at least one fixed point
which also means it is an equilibrium point. This does not show that
all equilibrium points are accessible from one strategy, however. It
just proves that there is an equilibrium point for each strategy, but
what he needs to prove is that all of the equilibrium points will
yield to the same step.

^{6}
This
makes it provable that any finite game has a

*symmetric*equilibrium point. Which means that it is not random, and it is not a sphere of points**that are related: but a curve of points that have a symmetric relationship. This is one half of the problem, because instead of a random series of points, he has shown that they lie on a curve. What he now needs to show is that there is one curve that dominates all of the others, if it exists. Remember it does not have to; there are unsolvable games, but if they are solvable then, they are solvable on a plane. This means that there are no strategies which have a myriad of unrelated points. Either they are solvable on a curve or they are not solvable at all.**
This means if you have a position which is
insoluble in any way, you do not have to look for any other examples.
Similarly, if you find a solution to a problem, that means that the
solution exists on any curve. This is unexpected, and the proof,
while simple, is hard to get through.

But
it has repercussions, because it means that to solve any problem
which has a solution, one has to start at the end and work backwards.
This is a general plan: start with the end.

But
then proves that for any game which is soluble, regardless of number
of players or their strategies, the same rule applies: if it is
solvable, work backwards. He called this the “Geometrical Form Of
Solutions”. And they had the property of Dominance and
Contradiction Methods”.

*5. More Than an Analogy*

At
this point, I am going to say that I fibbed, a little bit. I once
said that it was an analogy, but actually it is more than that. The
current space technology is one of two positional spaces, which
together add up to a single positional space. Unfortunately, I do not
have the time to recount the Higg's space side of the equation,
partially because the work has not been completely established.

^{7}
But the information space is the other road to this, so what I have
actually been doing is matching Gödel to amperage, Turing to
voltage, and Nash to ohmage. That means that this is not just a
mathematical concept, which some hints in Nash might have given you
the clue. One can show actual real meaning to this concept. Just as
AVO is related to current space, GTN is related to informational
space. Now you might think, so what? But that is where three values
not two enter the equation. Because there are three values, then the
left half of space, the one that current space is a part of, will
intersect with informational space. And in between these two spaces,
there is a space, which is one nor the other. In positional space
current is known, and Ohm is, generally, unknown. It is quite the
reverse for informational space: generally Nash is known, while Gödel
is, generally, unknown. There are exceptions to these, as with any
physical realities.

But
what this generally means is that the Nash Equilibrium is the
starting point, and then one calculates the Turing Machine to arrive
back at the Gödel Number. Of being a triangle, it can be from any
one of the three, but the Nash Equilibrium will generally be the
start, because it is the easiest one to find. And in science, if you
can find a quantity, then naturally that will be your starting point.
And the Nash Equilibrium, by joining the Gödel Number and the Turing
Machine, one being opaque, and the other not be decided upon, is the
only point where one has the advantage of being able to discern.

Thus
the normal process is to find out the Nash Equilibrium, just as with
the AVO the first part is to determine the voltage or the amperage,
since in current space those are usually the easiest points to
discern. In current space Ohm is not measurable, but amperage or
voltage is, by the fact that current is mainly about electrons, which
means you just need to place a voltmeter and measure. This is a
difference between current space and informational space. And it
might seem odd, and then dismissed.

Until
you realize that current space and information space really rely on
each other, and that means that there is a grouping of the two. So if
I divide current space from informational space, and join them, it is
obvious – in that way that mathematicians speak of obvious – that
there is a set which is neither true nor false. This, remember, is
from Gödel, that your two spaces are conjoined, there needs to be a
third space which is neither true nor false, but indeterminate. This
will lead you to Codd and Date, who are the informational engineers,
as opposed to the mathematicians. One of the important differences,
is the mathematician looks at the problem, solves it, and goes on.
The engineer actually has to build a working object. Think of it as a
fight between the scientists and the engineers over which is more
important: the discovery that it could work, or the actual working
object. This too, is part and parcel of the decisions you will make.

*6. The Greek Example*

Instead
of poker, I prefer politics. And after all, politics is a way of
people making choices which affect their lives. And it is a natural
laboratory for GTN. With real consequences. Let me take the current
turmoil in Greece as an example.

Do
not understand Greece? That is all right, you are with a great deal
of company, including people who really ought to know better. But
what if the papers today were a giant remark by a negotiator for the
Germans. Basically he said: "Why do not you just leave?"

And
that remark gave everyone who thought they had a clue, a real point
to start at the end and work backwards. It was just the smart people
who knew what was going on, it was the people who listened very
closely, which is a larger number than before. You just have to know
what the signs are, and use something called "Game Theory".

Game
theory was invented, to remind everyone, by Morgenstern and John von
Neumann, with von Neumann one of the finest mathematicians in the
20th century. The game that they started was littered with the best
of minds in the field, many of them great mathematicians themselves.
And three of the brightest are already mentioned: Gödel, Turing, and
Nash. What the three of them did was essentially invent the idea of
informational space, and, though separately, introduced the idea that
there was something like a current running through this. Later on,
engineers realized that it was not just theoretical, it was also
practical.

And
one of the greatest ideas came from Nash; that was, do not go
forward, go backwards. This way you will know what each participant
wants, and how best to conserve it. In this way you could show by
game theory what they would do, and what they would do by min maxing
their position, whatever their opponents did.

This
was not just theoretical, Nash showed it by using poker as an
example, but its main use is with real people and the choices that
they make. Once you look backwards, you will see that the logical
choices, if any, will be there. And sometimes they will not be there
as you would expect.

Take
for example: Greece. It might seem that there are only two players: a
Greek player, and a European player. From this viewpoint, the Greeks
want out, and the Europeans want in, that is of the euro. So the
Europeans punish the Greeks to bend them to their will. But there is
not two but three players in the game: the rich Greek people, the
poor Greek people, and the Europeans. The Europeans and the poor
Greek people both want for Greece to go on the drachma. But the rich
Greek people, who control all that is worth anything, want to stay,
so they can move out of the country whenever is worth moving. They
are the people who want to stay.

With
this in mind, it then becomes very simple to see why the condition of
the country is what it is. The rich Greek people are hiding money,
leaving exposed public goods to be taken; because after all they do
not care about public goods, only private ones. The poor Greek
people, who have some allotment of public goods, even if it is just
to get some to lose money, do not have this advantage. That is why
the rich Greekocrockes do not care about public goods, and are happy
to waste them in order to keep the private goods which they can
shuffle back and forward.

But
more than that, there are other rich elites wanting to stay on the
euro, whatever the cost is to their poor brethren. Such as Ireland.
All of these elites want to stay on the euro to have transportability
of their money, while the poor do not really care. At least not
enough to take the whipping from the bloodless Eurocrats.

So
next time you see the results of what is going on in Europe, realize
that the parliament is controlled by rich people, and the poor are
increasingly out on the street. And the rich Eurocrats do not have a
clue as to how to play the game. What they should do is go after the
rich of the country, not the poor. For example taxing money going out
of the country. Why? Because to them this is a sideshow, and what is
really important is not Greece, but Italy, Spain, and Germany.

^{8}^{}

But
that would leave the Eurocrats in the position of caring for ordinary
people. Which they would not rather do. I will leave as an exercise
how many people in Britain actually elect the government.

And
that is surprising, because we started out with a mathematical
concept, which seemed to have nothing to do with physical laws. But
here it is doing physical things, such as play computer games, which
some of you think I do not notice. But you would do well to notice
that I am up here, and you are down there, and there is an
informational space disparity, which does not exactly favor you. Just
noting that.

So
what have we discovered in this little talk? We discovered that there
is a numbering system, which is called the Gödel Numbers, based on
how primes are distributed. Then we moved to the Turing Machine, and
found that it could manipulate these numbers, and decide which of
them are useful in the present case. Finally we discussed Nash
Equilibrium, which was the resistance to manipulating Turing
machines. Finally we took this trio of mechanisms – let us call it
GTN – and found that there was a connection to current space.
Obviously this is only the beginning of the story, and whether one
works on it, uses it, or decides to run screaming from and only then
encounter it in an app, you are stuck with it as a feature of our
time. There will be other times, and some of you will go on to make
your mark: as these three young men did in their early 20s to early
30s.

But
remember, there is lot to think about, and the digestion will take
some time, and require more reading from you. You may find that there
is some mark that you can make in this lecture, and some future
lecturer may take a moment to recognize this. Realize you are getting
close to making your mark. And if I am alive, I will be very
interested in the direction you will take, because it is the new
which is exciting, and strange, and will take some time to get your
head around, as Someone by the name of Guth

^{9}did for me.
Footnotes

^{1}“Zermelo and the Early History of Game Theory”

^{2 }Actually: “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I.”

^{3}“Enhancing Patent Quality: Screening Out Low-Quality Patents And Trolling Litigation”, National Bureau of Economic Research Working Paper Number 20322

^{4}From the Antikythea mechanism, through Pascal's calculator, to Babbage's computer.

^{5}That is: MIT.

^{6}You can read up on Brouwer proof and a number of different variations on https://en.wikipedia.com/wiki/Brouwer_ fixed-point_theorem

^{7}At the same time my grandfather, Sterling Price Newberry, was actually involved in mining the Higgs Boson, though it was not called that, from electrons, were it exists, in to x-rays, where it does not exist; at GE.

^{8}With some fine work by Ian Welsh.

^{9}Cosmology at MIT.

Books

Aubin,
Jean-Pierre ,Mathematical Methods of Game and Economic Theory, Dover

Gödel,
Kurt, “On the Formally Undecidable Propositions of Principia
Mathematica and Related Systems”, Dover

Hofstadter,
Douglas H.,

*Gödel, Escher, Bach,*Basic Books
Kuhn,
Harold W.,

*Classics in Game Theory*, Princeton
Petzold,
Charles,

*The Annotated Turing: A Guided Tour Through Alan Turing's Historic Paper on Computability and the Turing Machine*
Turing,
AM, “On Computable Numbers, With an Application to the
Entscheidungsproblem”, Proceedings of the London Mathematical
Society 1936