Godel and platonism

I was reading in an old thread about how Godel was a platonist. Some people seemed quite confused by this because of godel’s incompletness theorm.

Incompletness shows that formal systems for proving theorms are incomplete; that there are certain truths that cannot be proven by the system but are undeniably true.

I am however not confused at all about why Godel should be a platonist. I think the confusion can come from 2 places a) they could be confused by what incompletness implies b) they could be confused by what a platonist believes.

I think point 1. is correct here but its important we phrase it a little more expictly

The platonic realm is complete but any attempt to represent it (in a formal system) as such is doomed to fail.

This is exactly what incompletness says. There are truths that are not contained within the theorem hood of a formal system. Plato’s world is not contained within a formal system.

But its slightly weirder than that. The truths or Godel sentences are sentences about a formal system written in the symbols of the system.

so

The truths are defined by the formal system.

But the formal system is defined by a human

The reason we know that the truth exists is because we(the human) can look outside the system and see the truth. Our logic still tells us that the statement is true.

So its not a case of logic(either human or plantonic) being incomplete at all. Its case of any attempt to formalise logic into symbol manipulation being is incomplete.

food for thought hopefully

a quick question: why should I not be a platonist??

Hi Fin666,

I was curious what you mean by the word complete, when you write, “The platonic realm is complete…”.

It appears to me that the Natural Numbers and the constructs of Multiplication and Addition are abstractions which have associated Ideals as part of the Platonic Realm. Can we not conclude that the Platonic Realm is a super system that includes this associated axiomatic system and is therefore either, incomplete and consistent, or complete and inconsistent? This would imply that Godel’s effort was futile.

I think that Occam’s razor was probably a death blow to Platonic Ideals in modern times. We don’t need Ideals if we understand what the abstractions are.

Ed,

Godels incompleteness theorem shows that any formal system that we create to capture all the truths about the natural numbers is either incomplete or inconsistant. Remember that the axioms and rules of these formal systems are created by humans(mathematicians).

Incompleteness does not state that the truths don’t exist. On the conatry it proves they do exist but are not provable by the formal system. So we could say the truth is part of a complete set of truths about natural numbers that do exist. So we can’t say these truths exist as an abstraction of the symbol manipulation but that they exist beyond this.

All I think is that any statement like ‘5 is prime’ is not a physical thing yet it in some sense is true in the same way ‘the kettle is black’ is true. But whereas the kettle’s blackness is a truth seen in the physical world at a location in time and space 5 being a prime does not exist in time and space. This is all I understand as the differance between phiscal and platonic existance.

‘5 is prime’ could be said to exist in a different ‘world’ because we can’t change this truth by manipulating this physical world whereas we can change the colour of the kettle.

I see that incompleteness only adds to this idea, that truths about natural numbers exist seperatly to the physical world, as it shows we can’t get to these ‘godel sentences’ via symbol manipulation alone.

No, The platonic realm contains the Truths alone; not the axiomatic system to reach the truths. We(mathematicains) create the axiomatic systems in an attempt to reach these truths by symbol manipulation.

I am far from an authority on the subject and may be completely wrong here, but it seems to me a wise man knows our human limits. I fear, those who think they can know absolute truth, are absolutely dangerous.
Therefore, a wise man would say, “this is all I know, but it is not all there is to know”.

I don’t think our brains are capable of perfect consciousness. That would be to know all there is to know. To be conscious of all of history and the motions of the whole universe, fully aware of the cause and effect of all things, even that which is not visionable to our eyes, nor audible to our ears.

Technolgical smartness, must not neglect wisdom. Leave room for what isn’t known.

Finn,

I think a careful reading of this article I found online would give a different perspective on Godel’s Understanding of his Theorum:

marxists.org/reference/subje … /godel.htm (especially part on Kant at the end)

In mathematics, we start with universal intuited principles. After some knowledge has been discovered, Godel claims Kant says, new intuitions grasp new principles that can be worked from. The new truths are intuited in a more traditional way rather than being posited as part of a system model – as I may have understood it elsewhere on these boards.

I don’t know if this is Platonist, except for intuiting knowledge instead of creating it. But from the article, it seems to be actually more Kantian.

Athena,

Welcome to the boards – I love your work so far. :smiley:

Don’t you think that with the realisation that we do not know everything there is also the firmness that we know something of the world and hope that we will discover more?

A wise post indeed. Despite you not knowing the ins and outs of godels theorm you seem to get the jist.

Godels incompletness theorm only refers to very formal, very human modes of thinking. It does not refer to Truth witha capital T.

That is the genius of Godel; he makes us realise that there is more to Truth than that which can be ‘proved’.

Hi Fin666,

Your comment “We(mathematicians) create …” prompts me to wonder if you are trained in Mathematics. If so I would like to offer a special welcome. I have a fond place in my heart for such people.

Now to less important matters.

While I can not answer the question that I posed rigorously in the affirmative, I do have some questions about your negative response.

Which constructs do you think lack corresponding Platonic Ideals? Some examples: Do you think that there is no such thing as a Platonic axiom, or Do you think that there is no such thing as Platonic Rules of Transformation? I think that axioms and Rules of Transformation in fact have corresponding Platonic Ideals. However, you may have something else in mind. If you do, could you please enumerate one or two examples?

I am also curious as to what realm of mathematics you might think that this discussion should be properly formatted. I haven’t quite figured this out!

Myself, I would suggest that Model Theory is the proper setting. In general I think that PM is a structure properly included in Model theory and the open question is whether or not there is a homomorphism into a substructure of the Platonic Realm. This is just a first swing at the ball though.

Thanks for your response.

Hi Athena,

I generally agree with your post.

As you have alluded, there are some areas of human endeavor where people could theoretically find the answer to any given question, (Euclidean geometry is both complete and consistent). However, as the systems become increasing complex, total knowledge (the ability to find an answer to a given question), even mathematical knowledge, becomes impossible.

Clearly, people that think they know it all are deluded!

But Ed3, as phrased in the Kantian answer I gave above by Godel, more complex systems find more answers. Some answers that cannot be found in Euclidean geometry (e.g., to create a seven-sided figure) might be solvable with more principles in a higher math – but no fewer answers are found. Is this right?

I’m actually physicist not a mathematician just about to start a postgrad coarse in theorectical physics. Thanks for your welcome.

Your questions are good ones. So I have to consider how to get some subtle points over of platonic ideas as I understand them.

I think the idea of an axiom and a rule of transfer are themselves platonic ideas. Then from these ideas of axioms and rules are created the idea of an axiomatic system. In a similar way to the idea of a triangle being created from straight lines and a 2-d plane. But these are just the ideas for general axiomatic systems or triangles not the PM or an equilateral triangle. Then I guess specific axioms and rules can be created by using other ideas.

Going back to incompleteness. If G is our godel sentence then we know G is true through our own concepts of truth and provability but not via the axiomatic system G is expressed in. So surely then the truth of any G, in any system, is accessible via ideas so the sentence ‘G is true’ exists in the platonic realm.

might not respond for a while because im moving house btw…

I just googled "platonic realm’ to get a better idea what people are talking about in this thread, and this thread is on the first page of google offerings. I did not see any sites on that page that provide a clear understanding of the platonic realm, and I know I should refer the early philosopers and improve my own knowledge, before I saying anything, but I am impatient.

I never associated Plato with math. His theory of form is not that helpful in understanding our physical reality, and became very problematic when it was all tangled up with religion.

Plato is the first to speak of the separation soul from body that was later adopted by Christianity. He strongly influence Christianity, at first indirectly, and later through his writings which were perserved by Arabs. “In the beginning is the word”, is not eqaul to a mathematical equation, and the ideas of perfectionism that followed Plato and Aristotle, are very problematic. During the period of Schalastism, the material learned men dealt with was abstract and metaphysical without being supplemented by knowledge of the concrete and physical.

When men like Kepler and Galileo began studying the physical world, they feared for their lives, because of the power of the church and the Scholastics, who rejected the idea of knowing anything by actually studying what is before us. We owe our knowledge of physical reality to men like Thales, Euclid, Pythagoras, Kelper, Galileo, Newton.

:blush: I do not have the information necessary to understanding why we should give much credit to Plato for our understanding of reality. Perhaps those who are better informed can explain?

Hi MRN,

The short answer, from the best of my knowledge, experience, and training is yes.

However this is an interesting topic to me; and I would like to elaborate, which will require some time for me to organize my thoughts. I will try to get back to you soon (maybe this weekend).

Thanks Ed

I would say Thales was a philosopher, and Euclid and Pythagoras were mathematicians (not very physical), like the Medieval Scholastics were to a large degree. Aristotle did a lot of observation on animals, so some might think of him as an early naturalist – “the master of those who know”. In a sense, both Platonists and mathematicians study forms separate from matter. A sign above the door of Plato’s Academy read: “Let no man enter here who does not know geometry.”

Kepler was a Lutheran seminarian, and Galileo’s daughter was a nun. I don’t think either had much of a problem with their church.

Fin666,
I do see the statement 5 as a prime as a description of something in the physical world. If we take ‘4 is not a prime’, for example, we can see how this is manifested in time and space in certain situations where a group of 4 members is divided into two equal parts, The same cannot be done with 5.
Just like the word ‘black’ is a representation of certain physical properties, so is ‘prime’.

You are right to question him, I think. Plato was very interested in Mathematics (allegedly he had the phrase ‘He who knows no mathematics shall not enter’ written above his door) but his interpreation of it’s importance was akwardly personal, peculiar to his own psychology, perhaps even pathology - he shifted the attention away from the physical necessities inherent in geometric representations of numbers, such as Pythagoras experimented with in very concrete ways, towards the unverifyable classification of experiences as metaphysically relating to these necessities. The tragedy is that geometry is the study of physical necessity, not of anything metaphysical.

Physical objects can be black but they can’t be prime. Concepts like members of groups are ideas in themselves. Of coarse platonic truths take effect in physical reality but the truth ‘5 is prime’ is not at a particular point in space or time.

You’re making the same point again. I still disagree. 5 is a decription of something physical, like the workd ‘kettle’ is, but even more so - it describes not just a form and use, but a set of physical necessities

Numbers are not metaphysical, not abstractions. One of the physical properties of 5 is it’s primeness.

I like this discussion, because it is really pushing me into a subject I really do not know. I tried to get a book on Pythagoras and was completely frustrated, but found other books that explain math as far more than computing numbers.

I especially like “A Beginner’s Guide to Constructing the Universe” by Michael S. Schneider. He explains the mathematical archtypes of nature, art and science and I don’t think they differ much from the archetypes of Gods and Goddesses.

The Greeks got their math from Egypt and possibly India. If the Egyptian concepts were not metaphysical and physical, I need someone to correct my thinking. The Eypgtians were dealing with math and the physcial world, and supernataural concepts, but evidently didn’t do the abstract thinking Plato and Pythagoras did? I really don’t understand the distinctions and explanation for crediting the Greeks with what they learned from the Egyptians. :frowning:

The power of the pharaoh was the belief that he held everything in order, and if he failed to do so, everything would fall into chaos and be destroyed. For a long period of time, Eygptian art was ordered by math. Measures such as the cubit were considered sacred. The numbers 1 through 10 are Monad, Dyad, Triad, Tetrad, Pentad, Hexad, Heptad, Octad, Ennead, Decad. These are not just numbers, but forces of nature.

When we get to the Mayan matrix, we are talking the forces that hold the universe together, and control the flow of what we call history. Their whole concept of langauge and life is one of motion, very different from our concept of “things” and materialistic language. I am not sure, but I think our materialism may blind us to truths of reality and distort our understanding of the ancients?

Very good discussion! You have got my perplexity perfectly. The number five is also the Pentad.

“It is a frequent assertion that the whole universe is manifestly completed and enclosed by the Decad, and seeded by the Monad, and it gains moverment thanks to the Dyad and life thanks to the Pentad.” Iamblichus

“The Pentad is particularly comprehensive of the natural phenomena of the universe.” Ismnlivhud

This quote needs to go with the patern in leaves, fruits, sand dollars, starfish, animal paw prints, the pentagram star and human form. I think we might be seriously underestimating the intelligence and consciousness of the ancients. The archetpyes of math are the forms and forces of nature, not just quantiatve symbols representing a number of things.

Hi MRN,

Concerning axiomatic systems and increasing complexity.

The following is a small historical review of some of the items that we are discussing. I wrote it as a jumping off point with the hopes that I could actually come to some reasonable conclusions. As you will notice I failed! However I did discover that my implication that Euclidean Geometry was a simple system compared to the Natural Numbers is probably unwarranted. They are simply different systems.

I am somewhat fearful that it will be redundant to you, as I view you to be well read. You can skip ahead to Body Here if you prefer.

There are many simple axiomatic systems, some examples of which are included in the book Godel Esher Bach and I assume the researchers in Formal languages have developed many others. Probably the most famous example of an axiomatic system is Euclidean Geometry. I believe that, until recently, Euclid’s Elements was the second highest selling book of all time.

Curiously enough, Godel offered a proof that Euclidean Geometry (technically the rigorously axiomatized form of Euclidean Geometry completed by David Hilbert) was both complete and consistent. Complete means that all true statements could be logically derived from the axioms or combinations of other statements (Theorems) that had previously been derived from the axioms. Consistent simply means that both proper mathematical statements A and Not A can be simultaneously true. This prompted Hilbert (significantly more important than most people realize), Russell and Whitehead to speculate that the Natural Numbers could be axiomatized and this system could be both complete and consistent. Additionally it should be noted that depending on the axioms in place there could be a number of different but consistent types of arithmetic.

But Godel in his famous theorem showed that an axiomatic approach to the Natural Numbers yielded a system that could not be both complete and consistent. Godel at the time did not realize it, but any larger system, such as the Whole Numbers, the Rational Numbers, and the Real Numbers would also have this same property.

So we have that Euclidean Geometry is complete and consistent, but an axiomatized Natural Numbers is not capable of being both complete and consistent.

Body Here:

I have wondered over the years what the consequences of adding and subtracting axioms from a given axiomatic system might be.

Very simple axiomatic systems appear to gain us very little information. For example if you assume that Descartes’ Cognito is true (a highly contentious statement), then you can only conclude that it is true. There is nothing else to discover.

With other more complex systems, we can even drop axioms and still have a meaningful system. Dropping the parallel postulate from Euclidean Geometry gives us the two standard non Euclidean Geometries.

It is clear that some systems can be greatly expanded with addional axioms. For example the Natural Numbers can be expanded to the Rationals. The Rationals can be expanded to the Algebraic (solutions to the polynomial equations can be added) and the Algebraic can be expanded to the Real by adding the transcendental numbers and finally the Complex Numbers can be added by adding i.

A confusing point about the Reals is that they are said to be complete. However the Reals are complete only in the technical sense that all sequences of Real Numbers, where the adjacent terms get arbitrarily close to one another (Cauchy sequences), actually converge to a Real Number. The Reals are not complete by Godel’s definition.

Aside from the obvious fact that useful axiomatic systems can give us interesting and useful tools by playing with the axioms (adding and subtracting axioms), I have been curious about the concept of adding a religious deity to the mix or at least developing a separate Ethical System with a presumption (axiom) of a religious deity.

I have also noted that different axiomatic systems can be combined. Mathematicians regularly study Analytic and Differential Geometry. I have never seen anyone explicitly mention the axioms used in these studies.

On how we develop axiomatic systems.

This is certainly getting into a contentious area. Do we have free will? Do we discover these systems? (Are they Platonic Ideals or Kantian synthetic a priori)? Do we invent systems? I will mention that you should take Godel with a grain of salt, because Godel did not create (if that is the right concept) any axiomatic systems. Like a good mathematician (or properly logician) he worked logically within a system.

Personally, I do not subscribe to any preexisting knowledge. For me the constructs that we develop are a process of abstraction drawn from the intellects’ (a genetic component should be considered) ability to differentiate and integrate. And of course the accumulation of data and experience.

No great insights. Sorry