The Paradox--A Warning?

The mathematicians David Hilbert (left) and Kurt Gödel (right), representing the classical view of mathematics (Hilbert) and upsetting new discoveries (Gödel), such as arithmetic’s incompleteness.  Both pictures are in the public domain.

"For with science and full passion
I have tried to answer all
Still it seems that in their fashion
All my reasons just appall'…

From the poem "Epic", by Rick Puetter

http://www.writerscafe.org/writing/rpuetter/298173/

 

 

The Paradox--A Warning?

     "…Ponderings on the reconciliation of science, evolution, and mathematical conundrums…what fools these mortals be..."

In the hotly contested battle over “Is man just a glorified machine, or is he something more?”, the struggle is fought at the highest levels by scholars well versed in what it realy means to be a machine.  Here subjects such as the Turing stopping problem, computability, Gödel’s theorem, and logic-system completeness, are foremost in the discussion¹.  Here the computationalists argue that all that man is, and can be, is a fancy machine governed by mathematical rules, while their opponents claim that there is much more to human intellect.  This battle is fought on many fronts: foundations of mathematics, physics, evolutionary biology, cognitive science, artificial intelligence, linguistics, and more².

What do I think? I think there are many reasons to believe that Man is just a machine (with qualifications--see below--because I think even the concepts of classical existence, mathematics, and what it means to be a machine are in question), and I can see no compelling reasons to believe differently, aside from the desire of some proponents to wish that it was otherwise.  There are strong mathematical reasons for my position.  There are physics arguments.  There are evolutionary arguments, and many others.  So just what does this mean?  To start an exploration of this question, let’s start with mathematics.  Why should we start here?  Well the best definitions of what it means to be a machine are mathematical, and mathematics is considered by most to be the highest form of human reasoning.  So, mathematics is tied up both in what it is to be a machine and in the question of what it might mean for the human intellect to be more than simply the output of a machine, i.e., does human intellect have more "god-like" qualities.  So here in the realm of mathematics, we have at least one well-developed discipline (but still not fully complete) in which we can discuss the plausibility of the existence of things beyond the finite.  So let's begin.

The most influential mathematician in the late 1800’s and early 1900’s was unarguably the German mathematician David Hilbert³.  While embracing the work of Georg Cantor⁴, Hilbert was a more classical mathematician.  Hilbert was famous for his “Twenty Three Problems,” which were all unsolved at the time.  One of Hilbert’s central goals for the future of mathematics was to prove the consistency⁵ of the axioms of arithmetic⁶, and when a man as great as this speaks, others rise to the challenge so that they can be listed amongst the notable in the history of mathematics.  Over the years, solutions to various of Hilbert’s problems have given rise to many prizes in mathematics, but Hilbert’s “greatest failing” was dealt by Kurt Gödel, who proved that mathematics was not the perfect, god-like structure that Hilbert believed it to be. Simply put, Gödel demonstrated that arithmetic was incomplete, and that there were mathematical statements in arithmetic that couldn't be proven to be either true or false.  And what does it really mean for something to be true when in principle that fact can't be demonstrated?

With such an apparent fundamental failure of mathematics, this leads us to ask the question, just what is this thing we call mathematics?  Mathematics has been studied both for its ability to provide a practical description of the world, such as in the disciplines of physics and engineering, and as an abstract subject, with a goal simply to learn the beauties and internal “truths” of mathematics.  But what is the fundamental motivation for mathematics?  It seems to me that without mathematics’ descriptive power, and its ability to predict the results of experiments, or the strengths of bridges and buildings, our interest in mathematics would fade and die.  In mathematics' application in physics and engineering, we get the sense, and develop an intuition, that mathematics contains truth, and by extension, we have developed the feeling that mathematics contains truths beyond human endeavor and has almost godly qualities.  Such ideas were first challenged in the 1930s by Gödel, although hints of the “bizarre”, and perhaps “unreal”, nature of mathematics, came many years before from Cantor, even if Cantor himself believed that these truths were revealed to him by God (see Wikipedia, http://en.wikipedia.org/wiki/Georg_Cantor, and references therein).

What do such “paradoxes”, or “problems”, really mean?  Should they be a warning to us that something is very wrong with the way we think?  Does mathematics contain truths that transcend Man, or is mathematics simply an imperfect creation, made by an imperfect Mankind, that simply has proven useful to describe parts of nature? It’s hard to know, isn’t it?  Yet we know that as a species, mankind is young.  Indeed, on cosmological time scales, we’ve just crawled up out of the muck.  And look at us.  No one would confuse us with a chimpanzee.  We’re something different.  We’d say we’re more advanced, but there is no doubt that we’re something else.  It took six million years of evolution to achieve this difference.  And what about a chicken?  Yes, for sure we’re more advanced than a chicken.  And by some estimates, that took 300 million years to advance to human form, starting from a distant, common ancestor.  So how does a chimp or a chicken think?  Are we humans more advanced?  Does a chicken’s concept of reality have any flaws?  And I don’t mean because it has the facts wrong.  I mean because it is in principle incapable of grasping how the Universe works.  Well, I’d guess yes, i.e., that a chicken’s ability to grasp reality has flaws.  So is mankind’s ability to grasp nature flawed, too?

Undeniably, our intellect has been honed by evolution.  After all, having a physical understanding that doesn’t align with reality is expensive: you die.  But how does evolution hone an understanding of quantum physics or set theory?  I think a misunderstanding of transfinite numbers is far from fatal from an evolutionary point of view.  So why would our mathematics and science be able to describe these things so well?  Or does it?  That is the real question.

Well, our science and mathematics has to have some alignment with physical reality.  After all, we can fly to the moon and we can split the atom.  But there probably is more to the Universe than what we can sense and experience, and our mathematics and knowledge will never describe things we can’t in principle know.  Or at least this description will remain untestable scientifically.  Is there any evidence that such things might exist, i.e., that there are things beyond our perception?  I think the answer is yes, and I think that one could point to dark matter as a candidate for something that lies on the border of what we can and cannot perceive.  There is 5 times more dark matter in the Universe than ordinary matter, but dark matter can't be seen because it doesn't interact with ordinary matter except through gravity.  So it is invisible.  We only know that it is there because on large scales (the sizes of galaxies and clusters of galaxies) we can see its gravitational effects.  But if dark-matter people existed, they could walk right through you and you'd never know.  Scientists are slowly finding out a bit more about dark matter (and the even more elusive "dark energy"), but are there other things out there that we can't detect?  Why wouldn't there be? So I’m afraid mankind’s knowledge, understanding, including mathematics, may forever be flawed, and perhaps only more advanced thinkers will be able “fix” the failings of mathematics.  Of course, the solution to such conundrums might simply be: why would an intelligent, thinking being even consider studying such “corrupted” mathematical systems?

While the idea that our human brains are flawed is deeply upsetting, my thoughts on the matter recently have lightened somewhat. While it is almost certainly true that Man is an evolutionary work in progress, perhaps we are only being careless in how we choose to do mathematics.  This is not a new thought.  The mathematical constructivists⁷ and finitists⁸ (a more extreme form of constructivism) have put forward more cautionary views regarding mathematics.  The finitists (and ultra-finitists) have taken the more extreme position, and do not admit the explosion of infinities into their mathematics allowed by the more traditional view.  (In the now mainstream view of set theory introduced by Cantor, there is an infinite set of different infinities, each one being larger than the previous one.) The ultra-finitists even discard the reality of extremely large finite numbers as they make no "physical" sense and have no substantiation in our Universe.  And what does this do?  Well not admitting a countable⁹ infinity, or writing down procedures that take a countable infinity of steps, removes the existence of larger infinities.  There are then no irrational numbers with an infinite, non-repeating decimal expansion, as there is no such thing.  Now these are complicated matters, and I am not an expert in this field, but you get the point.  These mathematicians require a higher bar to accept the mathematical existence of something, or should I say the “actual existence” of something, including things mathematical.  They have a strong aversion to things that might be considered “mind games”.

Now what might I mean (or ultra-finitists mean) as "mind games"?  As an example, let us take a look at a famous number such as Pi.  At the present time, Pi has been calculated to over 13 trillion decimal places.  But is this meaningful?  What if we wanted to scientifically test the voracity of this calculation, and make the biggest circle we could, and use the finest measurement apparatus to measure the ratio of the circumference to the diameter of the circle, i.e., the value of Pi.  If we were to create a circle that filled the Universe, and to measure the circumference and diameter with a ruler with graduation marks equal to the Planck length (the smallest possible size of space according to quantum mechanics), we could confirm the value of Pi to only 62 decimal places.  So what does it mean that we have calculated Pi to over 13 trillion decimal places, when in principle the finest circle we can construct is trillions of times coarser?  So Pi seems to have no meaning in our Universe.  So is Pi real or simply a mental construct?  In physics we use all sorts of mathematics, but we choose the mathematics that fits reality.  If it doesn't fit, we move on.  We find something else that works. We simply say that we haven't found the answer yet and note the limitations of our mathematics, understanding, measurements.  And we say, if appropriate, that this part of mathematics that we are using just doesn't match up to what we understand about reality.  No harm, no foul.  So perhaps we should just dismiss the idea of Pi as being non-physical.  It was perhaps useful when we didn't know that the Universe had a finite size, and when we didn't know that there was a smallest size of space.  But today we know otherwise. So there is nothing in the Universe that "needs" the number Pi.  We only have finite things in this Universe.

Another physics consideration strikes directly against the founding principles of mathematics, and it is ironical, then, that one of the most important applications of these physical principles will undoubtedly be in mathematical computations.  These are the ideas related to quantum entanglement, quantum computing, quantum decoherence, and related concepts.  The basis of all mathematics is the existence of classical truth.  One proposes mathematical axioms and then one proves theorems that are either true or false in an absolute sense.  This presupposes, of course, that this is realistic and possible, i.e., that there are things that definitely have one of two states, in this case either true or false.  Quantum mechanics, however, teaches that this is never true.  There is nothing in this Universe that has classical properties.  The simple, single electron system, for example, has a spin of 1/2 with the dipole, one might suppose, pointing either up or down. Quantum mechanics shows that this is not the case.  The electron is a superposition of states that are an equal mix of both up and down.  The process of measurement of the spin causes entanglement of the electron state with the measuring device, but there is still mixing of both up and down states in this entangled system.  While the interpretation of the measuring problem is still hotly debated, it seems that the eventual "settling down" of the electron state into either an up or down state after measurement is due to countless subtle interactions of the electron wave function with the environment around it (see Wikipedia's excellent article on Quantum decoherence, for example) and is in reality an illusion caused by reduction of the amplitude of the density matrix for the entangled terms through environmental interactions.  So, in fact, there are no classical objects in the Universe that definitely have absolute, given, set properties, and by extension there are no objects that are true or false, as would be required by mathematical systems.  The very quantum structure of Nature rules out the existence of such things.  And this is fundamental to the inner workings of the Universe.  If this were not the case, then if one tried to measure the spin of the electron, but this time decided to see if the spin was sideways and either to the right or to the left, you would get different results.  But you don't.  If you decided to measure the sideways spin you also get that it will be either right or left just like you previously determined that it was either up or down.  In order for these results to be the same, the electron has to be an equal mix of all states and not to have any one property before it interacts with the measurement device and multiple environment interactions. So in the real world there is nothing that is like a mathematical theorem that is either true or false, and if the world was based on such things (i.e., a mathematics-like system with preexisting truths) Nature wouldn't work the way we know that it does and the Universe would be a disaster.  So again, one is left wondering why mathematicians (most of them) have so much faith in the absolute reality of "mathematical truths".  There is no such corresponding system in the physical Universe.

And then there are the other mathematical “trouble makers”, i.e., self-referential, mathematical statements.  But are these true mathematical statements?  Should they be disallowed in the same way infinities are banned by the finitists?  Some call such statements “meta-mathematical” statements, i.e., statements about mathematics and not statements properly "within" mathematics, and it is these, self-referential, meta-mathematical statements that give rise to paradoxes such as Russell’s Paradox (Does the set of all those sets that do not contain themselves contain itself?).  Not admitting self-referential statements within mathematics also removes Gödel’s incompleteness result as his “Gödel sentence"¹⁰ would not be admitted.  Still, it is so easy to slip into such statements since natural language is full of such stuff, and it needs to be, so that we can talk about all possible topics.  But should all sorts of statements be allowed in mathematics, especially if they raise such conundrums?  And if such “slips” (and certainly many, if not most, mathematicians would not label them as such) are so easily made, perhaps there is something wrong with, or just something unevolved about, our thinking process.

Now why does this bother me?  I think, perhaps, that it is because I am a physicist rather than a mathematician.  Mathematicians believe in the reality of truth. I am not so sure.  I think that the concept of "absolute truth", i.e., a truth apart and independent of Man,  may be simply a human construct and may have no "real" existence.  But mathematicians generally believe in the "classical" reality of mathematics in the same sense that physicists speak of "classical" physics.  But the quantum world is a probabilistic world, or a world that bifurcates at every interaction. It is a world in which space and time were created and which didn't exist before.  It is a world in which effect can precede cause at least for short periods of time.  It is a world in which objects do not have properties until they are measured.  This is not a classical world, so do we even dare to think that there are classical things out there that have reality?  Physics teaches that everything we thought we knew before was wrong and things don't work in a classical way.  How is it, then, that mathematicians have so much faith in the Platonic existence of mathematics, transfinite numbers, etc., and especially to hold this belief in the face of so many conundrums and apparent paradoxes? The whole edifice has me shaking in my boots.

But perhaps this is a matter of religion when it comes right down to it, and relatively current studies seem to indicate that mathematicians are more religious than scientists as a whole, and that physicists and astronomers are currently the least religious group.  After all, if one believes in God, one has probably already embraced the idea that there is an existence beyond our physical Universe, and I don't mean in the sense of a multiverse as physicists might embrace, as the multiverse is just a collection of other physical Universes.  So here, in an outside reality, might be that classical, Platonic existence in which God and mathematics might reside.  However, there is no evidence that such a place exists, and belief in a classical, Platonic Universe is how all of physics began.  But experiment and hard trials have demonstrated that this is not how Nature works.  Nature is something very different and the classical picture is simply wrong.  So why then continue to believe in this illusion?  If a classical, Platonic reality actually does exist, we have no reason for believing so, and all the evidence from a century of physics seems to indicate that "reality" is very much different from this.  So if mathematics boils down simply to religion and belief, this is certainly different than the advertisements that mathematics contains the highest, known, "demonstrable" truths available to Man.

These are just the ponderings of a scientist trying to reconcile physics, biology, evolution, and mathematics, which gives rise, in my view, to an inescapable, very constructivist⁷, (ultra)finitistic⁸ view of mathematics and the world.  But remember, in principle, a true and complete understanding of this may simply be beyond me as a still evolving creature.  So unfortunately, I only have opinions.  My apologies.

 

 

©2015 Richard Puetter

All rights reserved

 

 

 

Notes

¹For those that are interested in this topic and who are unfamiliar with these terms, I urge you to look them up. The subject is not easy, but central to the issue.  That is why so many high-brow discussions are centered on this, and carried out by the world’s leading scientists, mathematicians, and philosophers.  Do not sweep the topic away because it is hard.  Bite the bullet and delve at least as far into it as you can easily go, then delve deeper.  It is worth the journey.

²There are many references that touch on the issues discussed here, but the interested reader might start with the article http://plato.stanford.edu/entries/computational-mind/ and the more technical, mathematical article http://www.ihmc.us/sandbox/groups/phayes/wiki/a3817/attachments/19f18/LaforteHayesFord.pdf?sessionID=b9fa9bb33fc036292df736e249fdf0a28d80d4f0.  While far from complete, these papers will give you a taste of the issues currently being argued.  As a scientist, I prefer the more mathematical papers.  But this is more than just a preference. The mathematical treatments show just how important careful definitions of terms are to what can and cannot be deduced, and one moves further and further away from careful definitions as one leaves the realm of mathematics.  And it is for this reason that I warn caution.  One of the difficulties in interpreting results is the use of loose definitions of terms.  So the reader has been warned.

³For an introduction to the work of this famous mathematician you can start with the Wikipedia article: http://en.wikipedia.org/wiki/David_Hilbert.

⁴Georg Cantor was another mathematician, whos ground-breaking work on set theory and transfinite numbers set the mathematical world on its ear--see the Wikipedia article http://en.wikipedia.org/wiki/Georg_Cantor for an introduction to his work.  Cantor set the stage for the mathematical developments central to the issues discussed here, and gave some of the first examples of deep mathematical conundrums, referred to by Poincaré as a “grave disease” infecting mathematics.

⁵Consistency of a mathematical system means that there is no theorem, T, in the system that can be proven to be both true and false.  If arithmetic were proven inconsistent, this would be a devastating shock to Hilbert, who strongly held the common belief that mathematics, in some sense, was a perfect system and the crown of human reasoning.

⁶According to Wikipedia, arithmetic is the oldest and most elementary form of mathematics and is a fundamental part of number theory.  It includes the operations of addition, subtraction (addition’s inverse), multiplication, and division (multiplication’s inverse), with identity elements for each operation (zero for addition and one for multiplication). Arithmetic is not the simplest mathematical system, as groups, for example, are simpler.  But it is one of the oldest fields of mathematical study.

⁷Constructivism is the philosophy of mathematics that asserts it is necessary to be able to construct a mathematical object before one can be certain of its existence.

⁸Finitism is a philosophy of mathematics that doesn’t accept the "Platonic" existence of numbers or mathematics, i.e.,  a "real" existence, but perhaps in a reality that transcends the Universe's physical reality .  The ultra-finitists even reject finite numbers that are so large that they have no possible substantiation in our physical Universe. 

 

⁹A “countable” infinity is the smallest infinity and is the cardinality of the integers and the rational numbers.

 

¹⁰Gödel’s “Gödel sentence” is a meta-mathematical statement, i.e., a mathematical statement about mathematics.  Let G stand for “the Gödel sentence” for a given mathematical system called T, which can be stated as follows: “G cannot be proved within the theory  T.”

I recently had a request for a list of math/science inspired writings that I've put on WritersCafe--see Great Aunt Astri's most recent review and my response.  I tried to repeat that list here as I thought I could list all the hyperlinks, but it won't save.  It reports that this article has too many links.  So here is the list again with just the URLs.  Sorry.

Math/Science-inspired writing:

Number theory http://www.writerscafe.org/writing/rpuetter/278847/

Epic http://www.writerscafe.org/writing/rpuetter/298173/

What creatures dream http://www.writerscafe.org/writing/rpuetter/314401/

Tiny Specks http://www.writerscafe.org/writing/rpuetter/626332/

Tiny Geometries http://www.writerscafe.org/writing/rpuetter/448007/

Hydrogen http://www.writerscafe.org/writing/rpuetter/490402/

Europa http://www.writerscafe.org/writing/rpuetter/430686/

Eris http://www.writerscafe.org/writing/rpuetter/1465155/

Man's Mind ' Reentrant http://www.writerscafe.org/writing/rpuetter/428478/

The Song of Trees http://www.writerscafe.org/writing/rpuetter/355140/

Looking to the sky http://www.writerscafe.org/writing/rpuetter/279463/

Author's Note

I recently had a request for a list of math/science inspired writings that I've put on WritersCafe--see Great Aunt Astri's most recent review and my response. I repeat my response here as you can only see the first few lines on responses to reviews and I thought this might be of some interest to some. So here is the response to that review.

Dear Astri,

I generally don't make open comments on the reviews of my writing, generally preferring to respond with private e-mail to thank the reviewer. However, this review seems to be a request for some additional scientific poetry, and I have already written a number of such pieces. In the hope that others might also enjoy a list of such writing, below is a list. I will also add this comment to the "Author's Note" because one can only see the first few lines in a comment so this list may go unnoticed unless I copy it there. So here is a list of math/science inspired work that I've written so far. I'm sure there will be others down the line.

Best regards,

Rick

Unfortunately, review comments do not allow for hyperlinks, so I've added some hyperlinks to the end of the Notes section of this essay. Nonetheless, here is a list:

Math/Science inspired writings by Rick Puetter on WritersCafe:

Number theory
Epic
What creatures dream
Tiny Specks
Tiny Geometries
Hydrogen
Europa
Eris
Man's Mind ' Reentrant
The Song of Trees
Looking to the sky

Reviews

[Marie]

This article fascinated me, as does all your work. I am dismal at math, and I shouldn't be. I was not properly grounded in school, but that's neither here nor there.

I realize that mathematics is a basic of life; it is actually the basis for my rhymed poetry, if I think of it that way.

[Roland Petrov]

Mankind's outlook is flawed but improving, though to what end I don't know. An interesting piece!

[Sheila Kline]

Dear Rick,

This is certainly the kind of discussion posed to the "think tanks" that abound in every field you mention - perhaps while sipping a glass of cognac, while pondering the why and wherefore of existence, one comes to certain conclusions which are in each person's presentation backed up by appropriate facts, etc....in the end we have similarities, and we have differences which shall perhaps always be present because of the functioning of the human mind, the "machine thinking part" as so many facts/factors enter into coming to said conclusions presented by each faction.

The study, the hypotheses, the proven facts all give us material to study, and it is fascinating to read and be given more material to "tackle" to come to a conclusion. I rather think the conclusion reached by each becomes somewhat personal based on our past experience, our view of philosophy, our interpretation of mathematical data. As you know from previous discussions, I am a proponent of Artificial Intelligence and think we are on the verge of the reality of it, if it is not already with us in the form of the "computer". The computer is said to be capable of only doing what a programmer instructs it to do, but is there evidence that "it" actually has a mind of its own? I think, note I say think as I do not have the facts to back up my assumption.

Therefore, if I believe in Artificial Intelligence, and that a computer might just be a function of such, then I would have to come to the conclusion that yes, man is a machine. I have just reviewed the many meanings of "machine" and I cannot argue that we do not fit the criteria. However, I must reserve a portion of my opinion to determine how the "spirit", the "soul" factors into the machine. Does every living thing, because it is a machine have a "spirit", a "soul"?

I am least of all a mathematician so to prove theories by mathematical analysis is not my "cog in the wheel", but I am willing to listen to arguments, facts, etc. In the end, I am intrigued but because of my religious beliefs, I have questions, but then again, if I believe that God created all things, or a higher power created all things, then a machine would certainly be included in that "ALL" category.

There is a rhythm to life, to the universe, to all things and this is supported by mathematical equations/data. When the rhythm is broken, we have a malfunction, much like a broken machine. And I must not forget the study and reality of robotics. Robots are almost human, and not far from being equal to us, but then again they do not have the ability to possess/display emotions. Perhaps that is for another discussion.

Well done, thought provoking, in fact mesmerizing my friend! I shall go to the links and attempt to glean some of the information to have a better understanding of your opinions/presenting facts.

Best Regards,
Sheila, "The Machine"?
I am, therefore I am........