Deductionandreality_patrick D Bangert

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Deduction and Reality:

Bridging Science, Religion and Metaphysics

Patrick D. Bangert

Table of Contents

Lecture 1: The Basis of Mathematical Logic...................................................................... 5

1.1 Introduction to the Course......................................................................................... 5

1.2 What Is Real? ............................................................................................................ 8

1.3 The Historical Roots of Logic................................................................................. 10

1.4 What Is Logic? ........................................................................................................ 14

1.5 Axiomatics .............................................................................................................. 16

1.6 The Structure of Mathematical Logic ..................................................................... 181.7 Truth and Falsehood................................................................................................ 20

1.8 Logical Operations and Relations ........................................................................... 22

1.9 Conclusions............................................................................................................. 27

1.10 Appendix: The Dog-Walking Ordinance.............................................................. 29

Lecture 3: The Structure of an Axiomatic System............................................................ 30

3.1 Aristotle and His Followers .................................................................................... 303.2 The Axiomatic System............................................................................................ 34

3.3 The Model Concept for an Axiomatic System........................................................ 36

3.4 The Equivalence of Two Axiomatic Systems......................................................... 383.5 Consistency ............................................................................................................. 41

3.6 Independence .......................................................................................................... 42

3.7 Completeness .......................................................................................................... 43

3.8 Categoricalness ....................................................................................................... 44

3.9 Euclid’s Geometry in the Plane .............................................................................. 46

3.10 Conclusions........................................................................................................... 48

Lecture 5: Deduction......................................................................................................... 48

5.1 Primitive Terms of the Logic .................................................................................. 495.2 Basic Definitions in the Logic ................................................................................ 50

5.3 Axioms of the Logic ............................................................................................... 52

5.4 Basic Theorems....................................................................................................... 57

5.5 Syllogism and Proof................................................................................................ 585.6 Developing Mathematics from Logic ..................................................................... 60

5.7 Conclusions............................................................................................................. 61

Lecture 7: The Limitations of the Deductive Method....................................................... 62

7.1 Review .................................................................................................................... 637.2 Induction ................................................................................................................. 65

7.3 Algorithmic Thinking?............................................................................................ 67

7.4 Recursion ................................................................................................................ 69

7.5 Hilbert’s Problems .................................................................................................. 717.6 Gödel’s theorems .................................................................................................... 72


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7.7 Conclusions............................................................................................................. 77

Lecture 10: General Relativity.......................................................................................... 78

10.1 Aristotle, Galileo and the Birth of Science ........................................................... 7910.2 The Newtonian Universe ...................................................................................... 81

10.3 What is Mass? ....................................................................................................... 8410.4 Albert Einstein’s Revolution................................................................................. 87

10.5 Special Relativity .................................................................................................. 90

10.6 General Relativity ................................................................................................. 92

10.7 The Nature of Space and Time ............................................................................. 9710.8 Conclusions........................................................................................................... 99

Lecture 12: Quantum Theory.......................................................................................... 101

12.1 A Discrete Space-time?....................................................................................... 10112.2 Postulates of Impotence ...................................................................................... 105

12.3 Unexplainable Experiments ................................................................................ 107

12.4 Quantum Theory ................................................................................................. 109

12.5 Uncertainty.......................................................................................................... 11112.6 Schrödinger’s Schizophrenic Cat........................................................................ 112

12.7 Quantum Mechanics as a Proof for the Existence of God .................................. 114

12.8 Causality and Determinism................................................................................. 115

12.9 Conclusion .......................................................................................................... 117Lecture 13: Quantum Mechanics and Ontology ............................................................. 117

13.1 The Infinite Potentiality of the Vacuum ............................................................. 117

13.2 Fundamental Particles Have a “Size” ................................................................. 120

13.3 The Ontological Interpretation of Quantum Mechanics ..................................... 12213.4 Active Information and Non-locality.................................................................. 123

13.5 The Uncertainty Principle ................................................................................... 126

13.6 The Classical Limit ............................................................................................. 12913.7 The Pauli Exclusion Principle............................................................................. 13113.8 Other Interpretations ........................................................................................... 132

13.9 Unity of the Laws................................................................................................ 135

13.10 Conclusions....................................................................................................... 136

Lecture 15: Tibetan Buddhism I ..................................................................................... 137

15.1 The Four Noble Truths........................................................................................ 140

15.2 The Wheel of Life ............................................................................................... 142

15.3 The Six Realms of Existence .............................................................................. 144

15.4 The Twelve Stages of Life 1: Ignorance ............................................................. 14715.5 The Twelve Stages of Life 2: Karma.................................................................. 148

15.6 The Twelve Stages of Life 3: Consciousness ..................................................... 14915.7 The Twelve Stages of Life 4: Name and Form................................................... 15015.8 The Twelve Stages of Life 5: Six Senses............................................................ 151

15.9 The Twelve Stages of Life 6: Contact ................................................................ 152

15.10 The Twelve Stages of Life 7: Feeling............................................................... 152

Lecture 17: Tibetan Buddhism II .................................................................................... 153

17.1 The Twelve Stages of Life 8: Attachment .......................................................... 153

17.2 The Twelve Stages of Life 9: Grasping .............................................................. 15417.3 The Twelve Stages of Life 10: Existence ........................................................... 156

17.4 The Twelve Stages of Life 11: Birth................................................................... 157

17.5 The Twelve Stages of Life 12: Death ................................................................. 157

17.6 The Western Wheel of Life................................................................................. 159


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17.7 The Tabula Smaragdina ...................................................................................... 161

17.8 The Eightfold Path .............................................................................................. 163

17.9 The Concept of Guru........................................................................................... 16617.10 Liberation .......................................................................................................... 168

17.11 Meditation ......................................................................................................... 16817.12 Conclusions....................................................................................................... 170

Lecture 19: Dignaga and Dharmakirti ............................................................................ 170

19.1 Know, Do, Expect!.............................................................................................. 170

19.2 The School of Dignaga ....................................................................................... 17319.3 Know the World.................................................................................................. 175

19.4 Reality is like an illusion..................................................................................... 176

19.5 Probabilistic Actions ........................................................................................... 17719.6 Properties of Objects ........................................................................................... 179

19.7 Deductions about Reality.................................................................................... 179

19.8 Conclusions......................................................................................................... 181

Lecture 21: The Nyaya-Bindu......................................................................................... 18221.1 Perception............................................................................................................ 183

21.2 Inference.............................................................................................................. 186

21.2 Syllogism ............................................................................................................ 189

21.3 The World ........................................................................................................... 19121.4 Knowledge .......................................................................................................... 193

21.5 Conclusions......................................................................................................... 194

Lecture 23: The Terms of the Nyaya-Bindu ................................................................... 196

23.1 Primitive Terms................................................................................................... 19723.2 Some Definitions................................................................................................. 198

23.3 Some Axioms...................................................................................................... 204

23.4 Conclusions......................................................................................................... 207Lecture 25: Deducing the Nyaya-Bindu ......................................................................... 207

25.1 The Three-Aspect Theorem ................................................................................ 208

25.2 The Hetuchakra ................................................................................................... 210

25.3 Negation .............................................................................................................. 212

25.4 Syllogism ............................................................................................................ 217

25.5 Fallacies .............................................................................................................. 218

25.6 Conclusions......................................................................................................... 220

Lecture 27: Extending the Theory beyond the Nyaya-Bindu ......................................... 221

27.1 Dialetic ................................................................................................................ 22127.2 Emptiness ............................................................................................................ 222

27.3 The Four Noble Truths........................................................................................ 22527.4 Mahayana Buddhism: Enlightenment entails compassion.................................. 22827.5 Mahayana Buddhism: Global selfishness is local altruism................................. 229

27.6 Circular Reasoning.............................................................................................. 230

27.7 Apoha .................................................................................................................. 230

27.8 Conclusions......................................................................................................... 230

Knowing the Instant Through Wisdom: A Systematization of the Nyaya-Bindu .......... 233

Abstract ....................................................................................................................... 2331. Technical Vocabulary and Introductory Remarks. ................................................. 234

2. Correspondence between Logic the Statements of the Nyaya-Bindu..................... 241

3. Construction of the Logic and Correspondence to Reality..................................... 245

4. Logical Systematization of the Nyaya-Bindu (Theorems) ..................................... 250


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Appendix: Nyaya-Bindu: A Short Treatise of Logic ...................................................... 263

A. Perception............................................................................................................... 264

B. Inference................................................................................................................. 265C. Syllogism................................................................................................................ 268


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Lecture 1: The Basis of Mathematical Log ic

1.1 Introduction to the Course

Welcome to the USC on “Deduction and Reality”. I am Patrick Bangert and he is Paul Crowther. This

course is called “Deduction and Reality: Bridging Science, Religion and Metaphysics.” So it’s a very

big course in the sense that it encompasses a great deal of things. So, first of all, I would like to preset

these words from Lao Tse: “The Tao that can be known is not the true Tao. The word that can be said

is not the true word.“ This is very true as concerns this course. Anything I will say is an approximation

to how things should be said and so I hope you will forgive me for explaining things as best as I can

which is not that well but I shall try. There is a website organized for this course: http://www.knot-

theory.org/usc. It doesn’t look very good but it contains content; that’s preferable. Here it is and you

can go and visit it at your leisure.

You probably want to know what this whole thing is about. Deduction and reality is a very big topic.

Reality; what is real? What is deduction? How can we approach reality by deduction? First of all, we

want to get an idea of something about reality which is not immediately obvious. Miyamoto Musashi

said: “Perceive those things which cannot be seen. … Do nothing which is of no use.”

What do we want to do? We can approach the universe in many ways. One of them is the rational

approach to the universe. The universe has many components. How can we approach some of them?

We perceive things in front of us. Are they real? How can we make some sort of deduction about

reality and about the universe as a whole? Totality as such, can we approach it in a rational fashion?

How far is it possible to approach these things in rational fashion? What are the limitations of this

approach?


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Now, of course, rationality has some sort of foundation from the mathematical angle, which I shall

present. Those foundations are logic and axiomatics. These two branches of mathematics are very

tightly interconnected but they are not the same. That’s one fo the points I want to bring out – logic as a

method and not as truth in itself. And then to what extent is it useful and helpful?

You might think mathematics is all not useful. Well, it is and I shall try to change your opinion about

that to some extent. Is it helpful as well as useful to some extent? Can we gain some insight into the

universe? Yes and no. It can only bring you so far and there is some step of belief involved which

relates to the axioms of the system and this belief you must generate on your own or not – as you

choose. So there are limitations to everything and certainly from the mathematical site. I am sure Paul

will tell us what the philosophers came up with.

Now as far as this is a course at a university there has to be, unfortunately, some component of

assessment. There will be lectures. Those are fun. You can attend them as you wish or not. As you can

see with the microphone in front of me we are recording the lectures. Those will be put on the internet

in the recorded fashion and the typed up fashion and so it is possible to be downloaded. One big reason

for this is that I know there are a lot of people interested in this and they are in many countries

distributed all round the world and they cannot be here. That’s one thing.

For assessment there will be three things: two essays and one exam. Two essays, one for each of us.

We have each produced a list of 8 topics twice over. So there is a first essay about the first half of the

course for each of us and a second essay about the second half, 8 topic for each of us, so 32 topics in

total. Those can be looked at on the website. For the first essay you choose whether you want to do one

on my list or his list and for the second essay you have to do an essay from the opposite list. So there is

one for each of us and you get fair distribution of marks from each of us. For the final exam, of course

it will be at the end of the semester and you need not worry about that now, there will be the same sort

of distribution there and so 50% of your grade will come from each of us.


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There will be some required reading for the course. For the philosophical part, Paul assures me that the

textbooks he has picked out are good books as a whole. For mathematics such a book does not exist.

This is rather unfortunate especially because of the legal situation of photocopying things. I am not

allowed to photocopy for you. You must do it on your own. Even though the things get copied the same

number of times it turns out to be legal if you do it and not if I do it. This is one of the examples where

logic can no longer get to the answers. The first bit of required reading is right here. I was allowed to

photocopy this because I wrote it. There is no copyright. You can pick a copy at the end please. It

represents the combination of lot of research that we did a couple of last years ago about what

mathematics is about. That gives you a 7 page definition of what it is. I hope its at least a little bit of

fun to read. Certainly I had that in mind when I wrote it. Last point, I have already mentioned, there is a

website it will contain the transcripts of the lectures. The website most importantly for the moment

includes the essay topic. [addresses Paul Crowther] Is there anything you want to say on the game plan

before I begin?

Paul Crowther: “Let me just say that Patrick will address the more formalised aspects of logic. On the

other hand, people who have tried to make sense of the universe as a whole have often made use of

deductive procedures in a less systematic way while still being systematic in another way. Einstein said

that the most incomprehensible thing about the universe is that it is so comprehensible. We are asking

the big questions and not just in the sense of the mystic ones but understanding the universe as a whole

and this is what the whole course is about. Patrick will give you logic and mathematics, I will give you

a particular tradition of philosophy, the so called rational philosophy. First, I will do some preparatory

work and then present arguments for the existence of God and other rational procedures and then I will

look at a series of particular rational philosophers.”

Yes, certainly one theme is the unreasonable effectiveness of mathematics in describing the universe.

So what will I do in his lecture? Paul has given you an idea on what his part is. In the first four lectures,

I want to present to you logic as a branch of mathematics. What is it? What can it do? What are it’s

limitations? In the next three lectures, I will go through some axioms that have been used before to

describe reality: particularly quantum mechanics and general relativity. Don’t be afraid I shall not use

any formulas for this, I shall just present to you what the underlying assumptions are. Then, in the final


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seven lectures, I will try to develop Tibetan Buddhism as an axiomatic system. This is somewhat new

and an experiment but not the transcendental part will be presented. It’s Buddhism as a philosophy of

reality; not the whole meditation and chanting parts of it. If you are interested in that, I am also very

interested in that and we can talk about it, but not here. We should that in our own private time. What I

shall do is present the philosophy as reality and that differs substantially from the western tradition in

some cases and in some other cases it very remarkably similar. So that will be presented and, in my

opinion, that philosophy is very well amenable to mathematical discussion and that’s why it is

presented.

1.2 What Is Real?

“[Reality is] a child which cannot survive without its nurse illusion.”

Sir Arthur Stanley Eddington

So let me begin with some background. Eddington is, if you don’t know him, a very famous physicist

of the 19th

century. He claims that reality needs illusion and of course this is one of the basic principles

of Buddhism.

What is real? That is a very crucial question for course on reality. We need to know what reality is?

Can we actually define reality? The complications with definition is that you define things in terms of

other things. And then those things have to be defined in terms of yet other different things. When do

you stop? That’s a crucial point that shall be addressed. You can approach reality in many ways and

pretty much every human being himself or herself will decide distinct method of reaching reality. There

are, in my opinion, two main extremes of doing things. One is the operational and other is the

transcendental. The operational method basically says we want things to be useful. We want to make

prediction on paper that in a way that can be tested in the laboratory. In other words, we want to make


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statements about perception. This is what natural sciences do. Then there is the transcendental approach

which, sort of says: I want to go to heaven after I die and I want to receive enlightenment and I have to

achieve this in some sort of intuitive approach which is certainly not based on logical scientific

measurable principles.

I think those are the main two methodologies by which one can approach reality. For me it will be the

main object of this course to discuss the operational approach by logic and mathematics and, starting

from a few little bits which we have to believe, how we can then develop the whole rest of the universe.

The transcendental approach is of course very much connected to the operational one, especially in the

Buddhist section that I will discuss later. First, because the actual practical Buddhist wants to gain

enlightment and this necessitates the transcendental meditational approach so it will come up many

times in the course. But the main thrust will be the operational deduction.

So what is reality? How can we achieve it? For example, is this table real? I can see it. I can touch it. If

I drop it, it hurts. So is it real? Fundamentally I have to make a decision whether I shall regard that

thing as real or not. It is not an a priori given, that it is real.

There is a famous skeptical argument of the “brain in a vat.” Imagine that what really exitst is not your

whole body but only your brain. In the laboratory, your brain is connected up to some machine. The

machine is very fancy. It can stimulate the brain exactly such a way that your sense perceptions, your

sight, your smell and all the other perceptions are controlled by the machine. It can therefore play a

kind of reality as a movie to you and because your sense perceptions are entirely controlled by the

machine, it is impossible for you to trust your sense perceptions alone to distinguish reality (that you

are a brain in a vat) from what is being played to you. Of course the machine doesn’t exist, yet it can be

conceived. So the thought is: “How can we distinguish it?” The answer is that we have to make a

decision. Are we going to regard sense input as real or not? This is one of the axioms.


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It will be one of the axioms in the Buddhist part that you must, or that we shall agree to, regard sense

perceptions as real. However what do you do if there are some defects? Say I am colorblind. I cannot

distinguish between green and red for example. Clearly in traffic that’s a problem. But it’s still ok

because most people are not colorblind. So we have a sort of vote. Is that thing green or not? I disagree

with you but there are dozens of you. So I will basically take your word for it. This democratic

approach to reality fails in some circumstances. Imagine that you are a schizophrenic person. Most of

you should have seen “A Beautiful Mind,” right? The main character in “A Beautiful Mind” saw

people that were not real. How is he supposed to know that these people are not real? He sees them; he

can touch them; he can speak to them; they answer him. Everybody else, with exception of his

imagined people, can not see these people. Of course those people see themselves. So we can have a

vote in the class. He is lecturing, so all his students vote “no, we don’t see them.” He votes we see them

and his extra people also vote “yes.” We have 30 against 4 and the class wins. Now lets say the

schizophrenic person imagines a big army full of people and he has a friend beside him. The friend

knows that they are not there but everyone in this huge collection sees each other of course. Now the

imagined collection of people votes that they see everyone else and they win in a democratic system.

Who shall we trust? Is my friend, who does not see everybody else, going to let all these votes count?

No because they are not there. But I see they are there. So we cannot really agree. At some point we

must make decision to believe.

1.3 The Historical Roots of Logic

So sense input is flawed. But we have to have some starting point and this is where the historical roots

of logic lie. Logic and deduction are, of course, very very old. However, the first actually recorded

system comes to us from a little piece of the Rhind papyrus, from Egypt about five thousand years ago

and it includes many mathematical formulae for calculating various things such as the volume of

pyramids. One of the problems it mentions is land surveying. This was a crucial problem for ancient

Egypt. If you remember your history, what happens in Nile Delta is that once a year the Nile brings a

big flood and deposits soil, which is rich in minerals for the plants, over the lands. Then the water

retreats again and because of the new fertile soil, Egypt became great civilization – they were able to

grow lots of food, more then they needed. They could trade the grain to obtain money and power. Now


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how did one farmer say: “This is my land.” Basically what they did was to use a post of wood or a

large stone to mark the boarders of their land. If a particularly big flood comes along, those marks are

lost. That’s a problem because then I can put my stone further away from where it was before thus

stealing land from my neighbor. That cannot be allowed. Of course Egypt is a big civilization. There

has to be some record of who owns how much. And then there has to be some method for going along

and measuring out this much and putting new markers on the ground so that I receive the 10 acres of

land that I used to own, not that I can somehow steal more or get less because someone else steals

something. So how to do this?

The most important point is that we agree. We must use one system of measurements of area.

Everybody involved, all the farmers and central administrators must agree to use one system of units so

that it it clear what an acre is. And everyone must agree on the rules of the measurement process. These

rules must be constructed in such a way that somebody else, later on, after the current people are dead,

can still apply the rules because my property gets inherited to my children who shall inherit it to their

children and the bureaucracy 200 years down the line must be able to follow the rules such that the 10

acres I own now shall remain 10 acres during my grand childrens’ life. So we must agree on the rule

and the results of the rule. The results of the rule must be uniform. The 10 acres of land here remain 10

acres over there and the rules must be able to be executed by people later on. This is the basic principle.

In ancient Egypt they used ropes with knots in them. From this knot to that knot is so units and you

measure it off. That is ok for the measurement of land.

But we want to discuss many more things. So we have to develop rules that are much more general.

Logic will get us much further than surveying land. It will be able to deduce many things from given

sets of assumption.

Aristotle constructed a system of logic( I prefer “a system of logic and not “the system of logic” and

that’s the main point later on) which contains three basic rules:


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1. Anything is itself.

2. Any statement has a truth value and this can be true or false.

3. No statement may have more than one truth value.

Rule 1 can be regarded as the definition of the verb “to be:” Anything, such as this table, is this table.

This statement is called the Law of Identity. It is a vacuous statement unless you consider it as the

definition of equality embodies by the verb “to be.”

Rule 2 is very important because statements could potentially be not true and false but something else.

One could, for instance, consider “nonsense” to be a valid truth value for sentences that do not conform

to the definitions of the terms. Alternatively, if we lack information, we could assign probabilities as

truth values.

Rule 3 is very crucial – it is the most important one of the three. Any statement has one definite truthvalue and that shall be very crucial in all systems of logic. Definiteness is the very essence of logical

deduction and if a statement could have two truth values at the same time then it would really mess the

system up.

There are many other system of logic possible. This is the first big message in this course. Logic is not

one thing – it is many. You have to decide which logic you are going to use and this decision isarbitrary. If, of course, what you want to do is lay down axioms about reality, about a physics

experiment and make deductions that are going to be the results of the future physics experiment, then

your choice of logic must be guided by these principles such as you want the results to be real

whatever that may mean.


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In generating other logics, you have to modify the starting points of Aristotle’s system. The second rule

is the one most frequently denied by people introducing more truth values than two. You must decide

which logic to use based on the application you wish to follow. The choice is essentially up to you.

For thousands of years, geometry, starting with the Greek school around Euclid and Pythagoras, was

regarded to be true in the sense of absolutely corresponding to reality. There is one geometry that

guides the universe and that’s it. Euclid wrote the book and that was the last word for a long time.

Surprisingly, from a modern point of view, Euclid’s work was used as a textbook ever since was it was

written up until about 50 years ago in high schools. This very popular book that enjoyed a monopoly on

the truth for a long time, was then shown to be wrong. Not only is it wrong in the claim that this

geometry corresponds to reality but it is also wrong in the sense that its theorems do not actually follow

from the assumptions it proposes. Mathematically it is a pretty bad book. The only reason that I

mention it here is because it has turned out to be so popular. Many mathematicians investigated

Euclid’s axioms and theorems got inspired by them, in particular by the errors in Euclid’s deductions. It

is fair to say that the investigation of the errors of Euclid has lead to a very large portion of modern

mathematics. The book had great influence not only on mathematics but also on philosophy.

Particularly Kant decided to build a whole system upon this and he regarded the geometrical axioms

that Euclid proposed as a priori truths.

What does a priori mean? It means: “True before you do anything.” It is supposed to be clear to you in

itself. Just by sitting on your chair and looking around you these truths are meant to be absolutely self-

evident. They do not need a proof because they are so obviously true. But they do not need to be

assumed either because they are true. This is what Kant claims of the axioms of geometry.

Through the work of many mathematicians, it has been possible to construct many geometries, in fact

infinitely many geometries that are all consistent, all different from each other and all different from


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Euclid’s one. So there are many geometries. If you then introduce the principle that you want this

geometry to correspond to the universe as a whole then you have to perform some sort of physics

experiment to see which one matches experimental evidence. Eddington actually did this experiment.

He went to a solar eclipse and measured the position of stars before and after the solar eclipse. The

observations revealed that the position changed. This was the first direct verification of Einstein’s

theory of relativity and it showed that the space of our universe is curved. Curved space acts in a way

that parallel lines may intersect which contradicts one of Euclid’s assumptions and Eddington thus

showed that we live in a non-Euclidean universe. This makes Kant’s claim of a priori truth of Euclid’s

axioms doubly wrong. Not only are they not a priori true but they are not true of the universe at all.

What happens is that the gravitational attraction of the sun is so strong because it has so much mass

that it can bend light. We consider the rays of light as straight lines according to the general theory of

relativity. During a solar eclipse, I can look at a star which is just beside the solar disk and measure its

position. Then I wait until the sun has moved and I measure the position again. These two

measurements are found to differ and thus we conclude that the presence of the sun in the path of the

light has curved the path of the light so as to fool the observer into thinking the star was elsewhere.

Even though Euclid’s geometry is not true of the universe, it is possible to have Euclid’s geometry as a

perfectly acceptable mathematical system. It has axioms and it has theorems that you can deduce from

the axioms if you clean up the mess that Euclid made and its perfectly fine. You just must not claim

that it is real.

1.4 What Is Logic?

The same thing happens with logic. Over thousands of years, Aristotle’s system of logic was regarded

to be the system similar to Euclid’s geometry. Only recently, in the middle of the last century, some

mathematicians got together and decided to construct other systems of logic based on different

assumptions. This is a crucial thing to realize: Logic itself is based on assumptions.


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Aristotle’s three rules are assumption of logic – they define what that particular logic is. If you deny

one of them, introduce a fourth or make some other fundamental change, then you construct a different

system and this new system is perfectly acceptable. It has axioms and theorems that you can deduce

from them. You just must not claim that any of these systems correspond to reality. If you do, then you

have to form some sort of experiment.

Most of you have probably read some stories of Sherlock Holmes, the famous detective of the 19th

century. He says that: “When you have eliminated all which is impossible, then whatever remains,

however improbable, must be the truth.” This is a very good statement of logic. In classical logic,

probabilities are not counted. Something is true or it is false. It is not more likely than something else.

If you have deduced that all these things are false then the negation of all these things must be true.

Logic is a method that begins from assumptions and obtains conclusions. Changing the method,

changes the conclusions. You must not claim that what logic produces is transcendent truth. The

assumptions are agreed to be true in the sense that the axioms have an associated truth value which is

equal to true. We must not say that the axioms are absolutely true of reality. Logic is then a method that

produces conclusions that are true in the same way that the assumptions are true – based on agreement,

not on reality. The conclusions clearly depend on the premises or assumptions. We lay down some

premises that we agree to hold up, then we get conclusions and the truth values of the conclusions

depend integrally on the truth values of the premises.

Judging the actual truth of the premises is not within logic. No logic, no systems of axioms must claim

that its axioms are true of reality. That is a judgment that is outside of the system. If you want to claim

that your assumptions are true then you are in natural sciences or in philosophy but not within logic,

not within mathematics. Within mathematics the assumptions are agreed upon. We shall not make

claims of reality. For the mathematician, “true” is an operational word. It is always relative to a given

set of assumptions that are agreed upon. The set of assumptions are like a point of view. From a

different point of view (different assumptions), the same statement may have opposite truth values. For

the mathematician this is fine as the two statements can not be compared being true relative to different


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basic rules. For the natural scientist this is a cause to abandon at least one of these two points of view as

unrealistic. If we claim that a statement is true of reality or of the universe, then we are making an

entirely different claim that has to be verified.

We must have premises. If we have no assumptions, we can do nothing. We take a statement that we

can prove mathematically from other statements. These statements are in turn proved by others.

Circular reason is not allowed as it is obviously senseless. So the circle must be broken somehow. It

can only be broken if we agree on one or more particular statements and accept them as basic to our

theory. Those are the premises of the system. We cannot do without premises.

1.5 Axiomatics

Axioms are formulated in terms of primitive terms. This is one more step towards being elementary. So

we have agreed now that we have to have premises. We have to have at least one statement that we

shall simply agree upon. This statement uses words or symbols. It has some content and we must also

agree what the objects that appear in the axiom are.

The terms used in stating the axioms are either primitive or defined in terms of primitives. Primitive

terms are terms which we use without definition, they are the analogue of axioms. I will give you an

example of an experiment that we did about a year and half ago of very large systems of axioms. The

English language has many words. All words in English language have definition in English. They are

given in a dictionary. If you do not know a word, you look it up. You read the definition to understand

what the words means. For this looking-up procedure to work, you need to know some words to begin

with. Otherwise you ca not read the definition and you do not understand the new word.


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You do not have to know all the words’ definitions because you can deduce some of them from

context. So knowing some percentage of the words in the definition that you find in the dictionary is

enough to understand the new word. The dictionary we used for this is Webster’s dictionary; it contains

over 99000 words different words. The words used in the definitions are themselves defined in the

dictionary. We ask: “What is the least number of primitives for the system?” In other words: “What is

the least number of words I must know in order to be able to learn all the other words?” If we knew no

words, we could not learn any word but we also do not need to know all words as the purpose of a

dictionary would then vanish, so this number is not trivial. It turns out that from the system of more

than 99000 English words you only need to know 244 in order to learn all the others. This is an

enormous reduction of complexity. As all words can be understood in terms of the basic 244 words, we

could conceivably communicate using just these few words, the others are merely abbreviations of

collections (definitions) of the basic words. These basic words, whose meaning has to be fixed outside

the system, are called primitive terms.

Using this approach, it is possible to reduce a large complicated system to a small simple one. The

original one can be restored by applying some basic rules to the small one. The small system is called

an axiomatization of the original one and the rules to obtain the original from its axiomatization is

called logic. Clearly there can be all sorts of different rules of transformation depending on the

application and that is why there are many different logics and no logic is better than another.

This is simplification process is what I want to do in the Buddhist part of this series. There are

documents that have many statements about reality that are heavily interdependent. It will be our goal

to resolve some of these interdepencies and thus make it look much simpler.

Primitive terms are those terms we must agree on the meaning of. Axioms are statements in terms of

these primitive terms and we again agree to uphold them. In mathematics an example of a primitive is

“set.” Set theory is a very basic branch of mathematics that deals with discussion of collections of

things. The word “set” can not be defined without introducing other words. In set theory, it is agreed


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that the word “set” shall be used and theorems proven about it but that its meaning is to be ascribed to

it outside of the system known as set theory.

Primitives are the substance in terms of which axioms are stated. One may view the axioms as relations

between the primitives, as rules by which we may modify collections of primitives or as definitions of

how the primitives act in a given situation but not as definitions of the primitives themselves. An axiom

might say: “A table has a four legs.” So table and leg are primitive terms of that system as well as some

number system that tells us what “four” denotes. The primitives gives us the substance of the theory

(the names of the objects which will appear) and the axioms tells about them.

The truth of axioms or the meaning of the premises is not to be questioned. We agree on them and that

is all. It is, of course, desirable to have few axioms as in the dictionary example. It is actually useful in

a very practical manner to say that these particular 244 words will enable you to learn the rest of the

language. In fact, language books that teach people English as a second language are actually based

upon lists of this kind.

Now of course we have in the back of our mind some theory that we want to obtain. We have to

construct primitives terms and axioms in such a way that we obtain what we want. Later on we shall

want to obtain Buddhist theory and we must find some primitive terms and axioms that would give it to

us.

1.6 The Structure of Mathematical Logic

Logic is used to systemize the whole endeavour. From a few basic things, we build up something large.

Together the primitives, assumption and logic will enable us to make conclusions; together they build


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the very foundations of mathematics and of any axiomatic system. Of course, it is possible to have

many distinct sets of axioms that will give rise to the same theory.

Logic, starting from the axioms, gets to conclusions. But how does it do this? There are some actions

that logic has to perform upon the axioms to reach conclusions. There are two such rules in classical or

Aristotelian logic: Modus ponens and Substitution.

Modus ponens tells us that if we are given that if some statement is true then another one is true and wehave somehow determined that the first one is tue then we are allowed to deduce the second one. This

looks like a vacuous statement but it is not because this statement if A then B is a claim in the system,

it is possibly an axiom or a theorem depending on previously assumed axioms. So it’s a claim. It is not

necessarily actually true. But within the system it is a claim and we shall treat it as if it were true for

argument’s sake. Then if we determine that A is true, then B is true. The statement “if A, then B”

simply connects the truth values of A and B, it makes no statement about the truth of A. If we find out

the truth of A by some means, then the truth of B is to be concluded. This is the rule of modus ponens.

If you have a long list of dependencies such if this then this then this then this and if the first one is

true, then you can deduce the last one. This is the essence of mathematical proof.

Substitution tells us that we may substitute a particular object for a general term in a logical deduction.

Suppose we have the following statement: For every table, the table has four legs. The rule of

substitution says that the general term “table” can be replaced by a reference to a particular individual

table we happen to consider and so will allow us to deduce that this table has four legs. From general

statements, we may deduce particular statements.

Those are the only two rules we need for classical logic. Put together some primitives, axioms and

these two rules of manipulating the axioms and they give us logic: mathematical logic. In the case of

Aristotle, the primitives are “equal,” “true,” “negation” and “and.” The axioms are: (1) A equals A, (2)


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A is true or false, (3) A and negation of A can not both be true where “false” is defined as the negation

of true and “or” is defined as the negation of the negation of A and the negation of B. Already you can

see this is getting complicated without the use of symbols. Introducing symbols, we can shorten the tale

a great deal. They may look scary at first but they allow us to write complicated arguments in a neat

way so that we may better see if we are making any logical errors. Discovering logical errors while

using a language familiar to us from daily life is very difficult indeed.

An important fact is that any statement is a theorem if and only if it is a tautology. Recall that a

theorem is a statement which follows from the axioms by applying the rules of logic. A tautology is

any statement that is true, no matter if the statements involved in it are true themselves. An example is

“A or the negation of A.” As the statement A is allowed to either be true or false, one of the two

statements A or the negation of it must be true and hence this claim is always true, no matter what truth

value A has.

Saying that something is something else if and only if some condition is satisfied is a typical

mathematical way of saying that what preceeds the “if and only if” is the same statement as that which

follows it. The phrase “if and only if” is typically abbreviated by “iff.” So we can write a mathematical

theorem like this: A iff B. This means that A is true if B is true and only if B is true, i.e. if B is false, A

is also false. Thus the truth value of A must be equal to that of B whatever it may be. Therefore we can

say that A and B are the same statement with respect to truth content.

1.7 Truth and Falsehood

“Whatsoever is descendent from the tree of cognition carries the dichotomy in it.”

The Sefer Zohar


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Logic enables us to deduce statements from premises while keeping the truth content of them constant.

Plato liked to believe that the universe is a poor reflection of the ideal universe in which all the ideas

that we use, such as tables, exist in the pure state. He claimed that this truth was absolute truth. Hilbert,

on the other hand, was a mathematician who made the opposite claim that mathematics is just

manipulating marks of ink on paper and has absolutely no connection whatsoever to reality. All you do

is you set up a few axioms, you mark down a few bits of ink on your paper and you make up a few

rules for transforming those marks of ink into other marks of ink. Applying the rules and seeing ever

more complex patterns of ink emerge is mathematics according to Hilbert. He was one of the best

mathematicians who ever lived and, of course, it is an extreme case but it deserves to be considered.

The possible truth values according to Aristotle are only two: truth and falsehood. Buddhism adds a

third: nonsense. Its possible to say true and false statements but also nonsense. Actually most things

according to Buddhism fall under the category of nonsense. The most popular statement that they

choose to exemplify nonsense is “a flower in the sky.”. A flower needs to be on the ground. It can not

be up in the sky and therefore that statement is nonsense. It is against the definition of what a flower is

to be in the sky.

Some logics add varying degress of may be. I can state that my car is on the parking lot. Bue I am not

exactly 100% sure. Somebody could have stolen from when I saw it last and I would not know. From

my point of view this statement has to be a probable statement with a certain, hopefully, high degree of

probability. Because of lack of information, this probability is necessarily less than 100%. That leads us

to fuzzy logic which is an engineering system that has been built quite recently and which incorporates

degrees of uncertainty because of lack of evidence.

Q: Is your car not definitely on the parking lot or off, why does it matter that you can not see it?


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That is the whole dispute. I am myself. You are a figment of my imagination. But even though I see

you, you might not exist. This is really quite difficult to determine. Is the statement true? What does

true mean? Is it true in an absolute way? Is there some God-like observer who determines what is true

and false and I am simple too stupid to find out? Or is only that true that I personally, as an individual,

determine to be true myself. This is of course the operational approach of engineering. Only those

things are true which I have observed to be true. I do not observe my car, therefore it is not necessarily

where I think it is. Even though some God-like being might exist who does see it. So this is a matter of

philosophy. You must decide whether you are going to allow the existence of absolute truth in the

absence of the possibility of verification or not.

Q: What exactly do you mean by “nonsense,” does it apply to the so-called “logical paradoxes?”

Nonsense refers to any statement which you cannot determine to have one definite truth value. A

statement essentially goes against the definition of the very thing, such as “flower in the sky.” It is

against the definition of a flower to exist without ground. Logical paradoxes will fall in that category.

1.8 Logical Operations and Relations

“For I am the first and the last. I am the honored and the scorned. I am the saint and the prostitute.”

Nag Hammadi

Now we start with the symbolism. If something is true then the negation of it is false. If something is

false then the negation of it is true.

TF

FT

-A A


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Now this table is called the truth table. A will be a symbol, a variable. It can be equal to any statement.

A is equal to “green cheese” is not allowed because it is not a statement which has a truth value. A

logical statement must make a claim. So A is equal to “there is green cheese on this table” is a valid

statement. It has to be a statement of a fact.

Any statement is either true or false according to Aristotle’s system. The negation is just the opposite.

We shall meet more complicated truth tables later on. But negation is actually the most complicated of

all logical operations. All the other ones are simple. Negation is like the complement of a set. Let us

say we have a set. The set is defined as the set of all people. With negation I want to say “not people,”

the set of all things that are not people. Taking the set of all things which are not in a given set is called

complementation. This already is complicated. The set of all people is well defined. There is a finite

number of people and the set of all people contains all of them. But what is not people? Not people is

everything minus a little bit, people. But what is everything? With respect to what am I to negate, from

what am I going to subtract people? With respect to what do we negate statements?

In the case of Aristotle, it is simple. There are two truth values and negation chooses the other one. As

soon as there are more than two things, we must be very careful with defining negation or

complementation, especially when it includes an infinite number of things such as everything.


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Russel’s paradox makes the problem with negation very apparent. We ask for sets of things and we ask

if the members of those sets can be members of themselves or not. For example the set of all teacups is

a set and clearly not itself a teacup and thus an example of a set which does not include itself. The set

of all non teacups is a set which is not a teacup and thus this is an example of set which does include

itself.

Being convinced that both self-inclusive and self-exclusive sets exist, we ask whether the set of all set

which are self-exclusive is itself self-inclusive. If it is self-inclusive then we get a contradiction because

the set has only such sets as members which are self-exclusive. So we think that it must be self-

exclusive. If it is, then it can not include itself but it was defined as the set of all sets which are self-

exclusive. So both possibilities lead to a contradiction and thus we have a paradox. This illustrates how

difficult the use of the operation of negation and the use of the word “all” can be. Much caution is

required in their use particularly if the set of “everything” with respect to which one negates is infinite.

I will not discuss this paradox further here. Much can be said and the resolution of Betrand Russel to

his own paradox is a three volume work. We will look at paradoxes to some extent in the exercises.

The resolution is that we agree, as an extra axiom of the system. We define that sets of objects are of

first “type.” Sets of sets of objects are of second type and so on. The axiom is that one must not

compare sets of different type. That’s an additional axiom of set theory that Russel introduced and with

that the paradox is resolved because the set of all sets is of higher type than the set itself.

FFF

FFT

FTF

TTT

A ? BB A


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So A and B are two different statements. They can be true or false. There are two possibilities each.

“And” is a relation between two statements and it is going to be defined to be true only if both

components are true. If any one of them is false, then the combination is false.

FFF

TFT

TTF

TTT

A ? BB A


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The relation “or” is slightly more complex from an everyday point of view. For example: “Do you

want your coffee with milk or sugar?” Is it acceptable to want both or not? Clearly it is ok to say I want

sugar. It is also clearly ok to say I want milk. But is it allowed in the system, if I view this on a strict

level, to say I want both milk and sugar. Well, that has to be agreed upon, if that is going to be allowed.

Is it an exclusive “or” which forces me to pick one but not both. It is an inclusive “or” which allows me

to choose both. In the truth table, you can see that we have defined an inclusive “or.”

TFTTFF

TFFTTF

FTTFFT

TFFTTT

- A ? -B A ? -B-B A ? B A


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Two statements may be connected in the form “if A, then B.” If something is true then something else

must be true also. If A is true then the whole claim is true if and only if B is also true. If A is false

however, the whole statement is true regardless of whether B is true or false.

Q: Why?

You must not ask why? This is a definition of the concept “if …, then …” This is one of the crucial

things. You must not question its truth. It is simply a definition. The only reason we introduce thearrow notation is to make writing simpler in the future. We do not need it as I am going to illustrate

below.

Consider negating B and combining it with A using the “and” relation. The results are shown in the

truth table. Lastly, negative this whole combination and we see that the truth values are exactly the

same as the ones for the arrow. As far as truth behaviour goes (and this is all we care about in logic),those two operations are identical. In this way we may use the last column as a definition for the

dependence relation “if …, then …” Later we shall find it convinient to use the dependence relation

directly and this is why it is notationally nice to introduce here. From a fundmental point of view, it is

unnecessary as it is the same as a certain combination of negations and intersections. This is the

equivalence of two syntactically distinct statements. Syntactically distinct but still equivalent, that is

important. You can have many statements that look different but are actually the same.

1.9 Conclusions

Logic itself is based on assumptions. We must assume something. There are many distinct logics of

which I have presented the one that Aristotle invented. The consistency is the most important

characteristic. Consistency is the third Aristotle rule that nothing may have two truth values at the same


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time. If that were allowed, then your whole reasoning indefinite. But everything else, you are allowed

to deny. Even the statement that a is equal to a, you are allowed to deny. So, to sum it up, a logical

theory must have these things. It must have primitive terms. It must have axioms that are only stated in

terms of primitive terms. It must have a logic. In other words, a method to transform axioms into

conclusions and the correspondence to reality is something all together different. It is something

outside the system.

Q: What is the difference between Kant’s a priori truths and Aristotle’s logic axioms as presented here?

The difference is in belief not in content. I state the system and I say it is an assumption. We shall

simply agree upon axioms and take it from there. Kant says that the axioms are actually

transcendentally true of reality itself. That is, it is not an assumption to him but a statement of a

property of reality. For me, it is an assumption without claim at the truth. The content of the statements

is exactly the same, they differ in the extent to which they are claimed true of reality.

Q: Is Zermelo-Fraenkel set theory the most eligible one to describe Russell’s paradox?

There are many set theories. All set theories are created equal and there are no pigs amongst them that

are more equal than others. Even within the Zermelo-Fraenkel system there is a lot of dispute about

some of the axioms, particularly the axiom of choice. No set theory is more applicable than another.Russell’s paradox occurs in almost all of them. That is to say in all set theories that have not been

patched by the theory of types which was Russell’s answer to his own paradox. In fact, Russell’s

paradox came up in the review of Frege’s book on set theory. So historically it can be viewed as a

direct attack on Frege’s particular version of set theory.


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1.10 Appendix: The Dog-Walking Ordinance

The following selection makes the importance of clear expression in logical conversation very

apparent. It is taken from “The Reader Over Your Shoulder: A Handbook for Writers of English Prose”

by Robert Graves and Alan Hodge (Random House, 1979).

“From the Minutes of a Borough Council Meeting:

Councillor Trafford took exception to the proposed notice at the entrance of South Park: ‘No dogs must

be brought to this Park except on a lead.’ He pointed out that this order would not prevent an owner

from releasing his pets, or pet, from a lead when once safely inside the Park.

The Chairman (Colonel Vine): What alternative wording would you propose, Councillor?

Councillor Trafford: ‘Dogs are not allowed in this Park without leads.’

Councillor Hogg: Mr. Chairman, I object. The order should be addressed to the owners, not to the

dogs.

Councillor Trafford: That is a nice point. Very well then: ‘Owners of dogs are not allowed in this Park

unless they keep them on leads.’

Councillor Hogg: Mr. Chairman, I object. Strictly speaking, this would prevent me as a dog-owner

from leaving my dog in the back-garden at home and walking with Mrs. Hogg across the Park.

Councillor Trafford: Mr. Chairman, I suggest that our legalistic friend be asked to redraft the notice

himself.

Councillor Hogg: Mr. Chairman, since Councillor Trafford finds it so difficult to improve on my

original wording, I accept. ‘Nobody without his dog on a lead is allowed in this Park.’

Councillor Trafford: Mr. Chairman, I object. Strictly speaking, this notice would prevent me, as a

citizen, who owns no dog, from walking in the Park without first acquiring one.


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Councillor Hogg (with some warmth): Very simply, then: ‘Dogs must be led in this Park.’

Councillor Trafford: Mr. Chairman, I object. This reads as if it were a general injunction to the

Borough to lead their dogs into the Park.

Councillor Hogg interposed a remark for which he was called to order; upon his withdrawing it, it was

directed to be expunged from the Minutes.

The Chairman: Councillor Trafford, Councillor Hogg has had three tries; you have had only two …

Councillor Trafford: ‘All dogs must be kept on leads in this Park.’

The Chairman: I see Councillor Hogg rising quite rightly to raise another objection. May I anticipate

him with another amendment: ‘All dogs in this Park must be kept on the lead.’

This draft was put to the vote and carried unanimously, with two abstentions.”

Lecture 3: The Structure of an Ax iomatic System

3.1 Aristotle and His Followers

“Common sense is not what you need if you’re going to find out anything worth knowing; it is

uncommon sense.”

Prof. Z as quoted by Eric Temple Bell

“Whatever is, is right.”


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Alexander Pope

This professor Z that has been quoted by Eric Temple Bell is a famous mathematician, who at the time

did not want to reveal his identity; that is why he is called professor Z. This is an opinion of very

qualified person. Bell himself is a famous mathematician. And it is true. This is what is going to apply

with logic because if you want to find out anything truly fundamental you have to modify, not the

super-structure, but the basics. So anything “common” you do not want – you want to modify the very

fundamental assumptions of reality as a whole. Then you can find something really worthwhile.

Alexnder Pope was an English poet and his claim was that “whatever is, is right”. So he defines reality

as true or truth as that which is real. We can view this quote on a few different levels. We can treat

either truth or reality as previously defined or basic and then take Pope’s statement as a definition of

the other concept, i.e. a definition of a synonym for the previously known word, or we can view both

concepts as previously known and Pope’s statement as a claim for a theorem. Everything depends upon

the basis of the system.

Atistotle and his followers used what were in the last lecture called conceptual truths, which are

sometimes called a priori truths. What they are meant to be is that they are supposed to be self-evident.

In another words they are not in need of any logical proof. Logical proof in the mathematical sense is

only possible if you make certain assumptions. Any theorem that you may prove is true in relation to

the assumption and you must not question the assumptions, of course. But this is not what is meant by

self-evident. They mean self-evident in the sense that they are clear regardless of anything that you

might throw at them. There are true without any experience. Even if you are alive without sensory

input: no eyes, no ears and so on, you should be able to regard these statement as true. So we do not

look at reality or hear something and then make deductions from this and come to the conclusions that

such and such statements about reality is true. Conceptual or a priori truths are true totally independent

of any experience that we might have.


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A conceptual truth is not derivable from premises, it is a premise which is regarded to be true in and of

itself. If it were derivable from premises it would be proven in terms of them, i.e. it would be a

theorem. So it is not deductive. It is not inductive either. Induction is when I make a few experiences

and generalize those to all possible experiences of a certain type that I might possibly have. That is

induction but it is still based on some experience. I have to have made a certain few experiences before

I can induct to the general one. The conceptual truth is completely self-evident. Out of innate ideas this

statement is regarded to be true. Aristotle and his tradition have had a very profound influence on

mathematics as a whole. From the time Aristotle wrote it until approximately 1930 when a few

Hungarians came up with some more logics, Aristotle’s logic was regarded, even by mathematicians to

be “the only single logic” possible. This is a pretty major statement now that we know that there are

many logics, in fact an infinite number of logics.

For a long time and during the lifetime of most of the famous mathematicians you hear about, this was

not at all regarded to be such. They really thought that the logic of Aristotle was ‘the’ single logic and

it was self-evidently true. With the advent of questioning logic, which came together with questioning

geometry, several schools of thought formed within mathematics. The mathematicians began to really

think about what truth actually is. For millenium we thought that the Aristotle was the answer. Now we

know that he is one answer among many. Somehow we must choose intelligently between them in

some fashion. We cannot choose logically because of course that is the very thing that we are trying to

choose.

There are two main schools of thought in mathematics which differ very extremely. The intuitionist

mathematicians are different from the philosophers who call themselves intuitionists. The intuitional

mathematicians are basically Platonists. They believe that a mathematical idea actually exists in a super

reality of ideas and exists in a pure state independently of us who are thinking about it and the way we

formulate it. The way we think about it is a poor man’s version of the real idea that lives in the ideal

space. To sum it up, an intuitional mathematician, when he proves a theorem, makes a discovery. He

discovers something, which is true, which has already existed in this realm of ideas for all eternity until

this mathematician simply manages to discover it. In that sense mathematics is a science that discovers

things.


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On the other hand you have the formalist mathematician. This is David Hilbert’s school which

basically says that the axioms and the rules to transform axioms are nothing more than marks of ink on

paper. They have no meaning whtsoever. They have no relation to reality whatsoever. Questioning

their truth or their meaning is entirely senseless. We simply modify marks of ink into other marks of

ink and if you attach a meaning to this or if you claim any of this is true, you are no longer a

mathematician – you have become a natural scientist or philosopher. From this point of view the

mathematician can never discover anything. He can only invent things and the only question of any

relevance is if such an invention is in any way useful. The question “is it true?” is totally irrelevant to

the formalist from the point of view of mathematics. If you want to apply any of these inventedmathematical concepts to the natural sciences, the question “is it true?” becomes very relevant. But as

far as mathematics in and of itself is concerned, for the formalist, it is invention entirely and, for the

intuitionist, discovery entirely. To this day, there exist these two schools and several others of medium

flavor and the mathematician has to choose to which school he or she belongs. It is up to you.

Personally I belong to the formalist school. I think that if you demand that things are true about the

reality, you are no longer doing mathematics, you have suddenly jumped to physics or to another

natural science. I do not believe that mathematical ideas exists in an ideal state. So I would say any

discovery I have made is not a discovery at all, it is an invention. I dream it up and it has its own

reality, within the minds of those who read about it. There are many others who are of the intuitional

variety. There can be many fights between these two schools but because they differ in fundamental

assumptions, it is not really fair to compare the two. That statement in itself is a formalists statement of

course because to the intuitionist the fundamental ideas which form the assumptions exists in a pure

state and so should be capable of comparison. This is why you can fight but you must not become

angry because the two are just different starting points for developing mathematics.

In the last lecture, four concepts were mentioned that Aristotle made to describe his metaphysics and a

long explnation was given of each. Form the point of view of mthematics we would have to regard

these four concepts as “primitive terms.” They essentially incapable of a definition within the system.

A definition can be given but that definition is necessarily outside of the system and so this definition

becomes a mere motivation for introducing that new term. We were given these terms, such as


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substance and cause, and we understand to some extent what those mean. But this understanding is not

within the system. They are meta or outside the system, hence the term “metaphysics”.

3.2 The Axiomatic System

“Through natural things, we obtain physical powers; through abstract, mathematical and heavenly

things, we obtain transcendent powers.”

Heinrich Cornelius Agrippa von Nettesheim (1510)

Agrippa already claimed that mathematical thinking and mathematical inventions are more important

than natural sciences. You can have some insight into the nature of reality that goes beyond sense

perception. Physics and chemistry and such of course deal only with things that we can perceive with

the senses and abstract mathematical and heavenly things can go beyond that and that is why they are,

in some sense, better. We shall meet the alchemists again later on, for the moment that is all I want to

say about them.

We know that we must have primitive terms from our previous discussions. Let us me make a

distinction. Primitives can come in three flavours. First of all they may be objects. Those are, if you

will, proper nouns. I can say “table” or “set” for example. Those are objects in and of themselves. They

can have a meaning. Of course the meaning is not within the system. But they are to be regarded as

things. Then there can be relations. These are also primitive terms. The relation “between,” for

example. In geometry, we have the relation “between” applied to three points. It is not, however, an

object. It requires objects to make sense. The third kind of primitive is an operation, for example,

negation. While a relation connects several objects, an operation changes one object into another one.

The operation of negation changes the statement A into the statement “not A.” Those are the three

kinds of primitive terms and, in general, we need all three of them to construct an axiomatic theory. In


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other words, to make meaningful conclusions we have to have objects to talk about, a method to

change an object into another and a method to connect several objects together.

Q: Can you operate upon an operation?

No. Once you have operated once, the result is a changed object. You may operate again but then this

operation is upon the changed object and not the operation. You may think that the usual law of double

negation, i.e. “not not A is the same as A” is an operation upon an operation but it is not. First youoperate upon A with negation and get the changed object “not A” then you operate again to obtain “not

not A.” It just so happens, because of the way “not” was defined that negating twice brings us back to

the starting point.

The axioms, which are the basic statements that we agree to be true for the system, are formulated only

using these primitives. So you have to somehow make sentences out of the primitives to form theaxioms. As an analogy consider the the English language. Objects are nouns, relations are verbs and

pronouns and operations are adjectives and adverbs. The axioms here are sentences that we agree are to

hold. As we know from grammar, we need nouns, verbs, pronouns, adjectives and adverbs to make

meaningful sentences.

All the words that appear in the axioms must be either primitives or other words that have been definedin terms of the primitives. The primitives themselves have no definition. As we discussed in my last

lecture with the dictionary example, every word in the English language has a definition in terms of the

others words in the English language. But you must, at some point, pick a set of words that we simply

agree are known to be able to learn the others. Those few are incapable of a definition within the

system unless you allow circularity. Of course, circularity shall not be allowed because that would

remove any meaning from the system whatsoever. You must have a starting point. So primitives in

Aristotle’s case, for example substance and cause, are simply words. If we agree that they mean


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something we are outside of a logical system based on these words. If we define them in terms of other

words, some of these other words must be basic terms. The axioms that we formulate in terms of these

basic primitives are not conceptual truths. They are simply statements that we agree to be true; true as

parts of the system, not true absolutely or true of reality. Many mathematicians have regarded such

axioms to be conceptual truths. A long time ago, Euclid formulated his famous geometrical axioms. He

regarded them as conceptual truths and for thousands of years mathematicians followed along until

very recently this was shown to be wrong.

So axioms are regarded to be true, agreed to be true but they are not self-evidently true. In such a way

the axioms are like a rule of a game. Chess and the rules of chess are not true; neither are they false.

You simply agree to uphold them. If you cheat, if you disobey any rules of chess, you suddenly play a

different game. The new game you play is not true or false but it is contradictory to the original rules

laid down for the game called “chess.” If you disobey the rules of a game, you are playing a different

game; truth does not come into it. This brings us to an important point about “fallacies.” Many authors,

particularly philosophical authors, spend a great deal of time talking about fallacies – obtaining

conclusions that do not follow from the axioms. A fallacy is a claim which does not follow from the

given axioms. It is thus the result of disobeying a rule of the game which the axiomatic theory

embodies. So from the mathematician’s point of view a fallacy is simply ignoring or applying

incorrectly one of the rules of the game. That is all I want to say about them but in philosophical

textbooks you will find a lengthy discussions on various types of fallacy.

Axiomatic systems can have several properties. The five most important ones will be discussed here:

Equivalence, Consistency, Independence, Completeness and Categoricalness. The last two are really

the same property and equivalence is not a property of a single axiom system, it is a relation between

two of them.

3.3 The Model Concept for an Axiomatic System


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The formalist views axioms as sentences in terms of primitive terms and not as conceptual truths and

logic as a method to transform these statements into other statements called theorems. To the formalist,

this transformation process is mathematics.

If we want to talk about the axioms, we are doing “metamathematics,” i.e. going beyond the system

that is mathematics. You do metamathematics if you talk about primitives and axioms using normal

language. For example, if you write a book about it or explain concepts to others. When you read a

textbook about mathematics, there are lots of explanations about what the technical terms mean and

how the proofs are constructed. All of this is metamathematics. Only the actual symbols, formulae and

the derivations within the proofs are mathematics. All the rest is explanation. Of course the explanation

is crucial for understanding and communication but it is important to differentiate between the

substance and the presentation.

In theory you should not need all these explanations. The explanations are just there to expediate the

learning process of mathematics. It is not essential. That is why it is metamathematics. It gives these

basic terms, the primitives and the axioms some meaning and the axioms some truth. And this is called

a model.

A model is some object for which the primitives have a meaning and the axioms are true. An example

is Euclid’s geometry. Euclid’s geometry is an axiomatic system that has primitives such as “point�

Deductionandreality_patrick D Bangert

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