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.

[FGFRANKLIN]

I'm not a mathematician. I'm a philosopher of magic. I believe there's all sorts of magic at work everywhere, all around us, but the skeptics and the cynics want to drag it out into the harsh halogen lights of reality so they can park their ideas under it in relative safety. But stranger still, the more they pull, the more the magic pulls back and the facts tumble and we're back at square one. I enjoyed recently reading about how the JWST has upended the accepted scientific theory of the universe and everything science thought they knew has had to go back to the drawing board. But it's not out of a desire to see science fail but about getting some acknowledgement that nothing is as cut and dried as anyone believes it to be. But science is about proven facts, not a general consensus of opinion and some folks seem to have forgotten that. Theoretical physics is an interesting branch of science because it poses a lot of what if questions hoping that if the right question is asked it will produce the right answer. But the right answer once may not always be the right answer if there are variables in play. I'm not someone who needs all the answers. I like a little mystery here and there. It keeps things interesting and things get more interesting every day from my perspective. I enjoyed the read.

[Rick Puetter]

Hi Fabian, hanks for your review. As you know, I generally do not reply openly to reviews of my work, preferring to reply privately, but your review brings up so many interesting topics, that I though others might also be interested in my reply.

And yeah, I think I have a pretty good idea of your take on these topics. I get the perspective of there being a "magic" to life. However, I going to stick up for the scientific perspective since the accumulation of knowledge and understanding is quite considerable these days, and this demands the respect it is d
ue. Now be assured, I certainly would never claim that we know everything. We certainly don't, and the vast majority of understanding the Universe lies beyond our reach. But still there is quite a bit that we do know.

And I'm glad you are following the fascinating information coming out of JWST. That, of course, is my exact field of expertise, and I proposed to build the JWST instruments, but my colleague, Dr. Barbara Jones and I, lots this NASA competition to our friends and colleagues, Drs Marsha and George Reike, at the University of Arizona. But I, too, have been closely following the JWST and enjoying it's scientific triumphs vicariously.

Anyway, let me make some comments on what JWST is discovering about early stars and galaxies. The popular press loves to cast the new discovering in the exciting headline: "Previous Theories Proven Wrong", etc., because that sells newspapers and more modernly advertising on websites. But the new JWST discoveries do little to weaken current ideas about the Big Bang, which is now such a secure theory regarding the general structure, that little I've seen so far can shake this very much. What it does challenge is our ideas of how early stars and galaxies form, and that's what scientists are now struggling to reformulate and understand. But the facts are he facts. Galaxies seem to form much earlier that we thought. One potential reason is there are more primordial black holes than were expected, and these are providing "condensation points" around which galaxies can more easily and quickly form. Ideas such as this are now being studied, and such processes can go on without upseting the grand structure of the Big Bang, which now correctly and independently predicts dozens of properties of the Universe.

So for sure, there are still surprises in the makeup of the Universe, and JWST is blazing the trail in discovery these new aspects of cosmology. This was to be expected. That's what happens when you deploy a new set of instruments on JWST that are millions of times more sensitive than the Hubble Space Telescope.

[FGFRANKLIN]

I love the JWST and the new facts that are coming from it. There is no denying facts. But too many accepted scientific theories are treated as facts and that should not be acceptable to science. It seems so difficult for science to say, "We don't know." But without saying that, it doesn't leave much to investigate. I don't think the "Big Bang" "theory" is fact. There are just so many variables at play in the universal scheme of things that our miniscule knowledge is like a single drop of water in an ever expanding ocean.

[Rick Puetter]

And that is just my point. And we've been studying cosmology at a serious scientific level about 100 years. Further, you are right, there are so many possibilities regarding how the Universe formed, it is so highly unlikely to get the right answer. But there is a correlary to this. Since any given answer is so unlikely that any one given theory woud get several dozens details that could take any value, and get all of them right to better that 1% accuracy? But that is what the Big Bang theory currently does. And this is all completely surprising to astronomers. At every launch of a new mission, astronomers were betting that the results would cause us to abandon the Big Bang theory. But the exact opposite has been happening for more than 50 years and some dozen of experiments. It has turned into a dazziliing victory. Yeah know that when you predict two dozen variable to 1% precision, the odds that this would happen at random is astronomicaly small. And some of these tests were suggested by the Big Bang theory itself since there is no competing cosmology that would even hint on there being such structure. For example, if there was a Big Bang you would expect gravitational oscillaions and vibrations in the fabric of space time. So several experiments went out to measure these oscillations. Without the Big Bang, the expected result was to see nothing. When measured, however, there was a highly detailed structure that complelely is as predicted by the Big Bang--see figure below. So today, the Big Bang theory is the most secure theories in all of physics, and any doubt in the Big Bang just indicates you are ignorant of the overwhelming facts, and you''d need to come up with a new theory that explains all the predictive triumph of the Big Bang when alternative theories have no known mechanism to reproduce these results.

/Users/richardpuetter/Desktop/Screenshot 2023-09-02 at 11.49.19 AM.png

[Rick Puetter]

https://universeadventure.org/big_bang/popups/cmb-dtrh-powerspectrum.htm

[FGFRANKLIN]

We still have to question, if the universe came from a "Big Bang" what exploded and where did it come from? Some say the universe formed from a mass about the size of an egg. The creationist theorists say the universe was created by an implosion rather that an explosion, that the cosmic dust of the universe was gathered together in an instant and then released at what they call the beginning of time. Perhaps the "Big Bang" isn't or wasn't exactly what we think it was. We may never know but I'm encouraged by the amounts of new data being discovered and studied every day by science. I love studying the stars.

[Rick Puetter]

You are absolutely correct. There will never be an ultimate resolution of why, what was there before and if there was nothing there before, how can that happen, why, and how. Final answers are always beyond the grasp of finite man. But it is equally amazing to me that we can have some answers to these questions, e.g., when did our Universe start. How were the primordial elements formed. How smooth is the Universe and why. What is the Universe made up, eg., dark matter, dark energy. What gives rise to ununiformity in the Cosmic Microwave Background. How are we so luck to know things on this level. The whole thing is a marvel.

[FGFRANKLIN]

Dark Matter, Dark Energy...sounds like Darth Vader's neighborhood. ;) Black holes, wormholes, rifts in the time space continuum...and the Large Hadron Collider at CERN finding all sorts of interesting things that mankind never knew but the parameters keep expanding. We need marvel and awe which is why I propound the argument for magic at work. Arthur C. Clarke once famously wrote, "Magic's just science that we don't understand yet."

[Robert Haigh]

A fascinating subject, and one we may never fully understand. I play chess, which is certainly a mathematical game in many ways, and mathematicians tend to make good chess players. However, there is also the scope for fantasy and risk-taking in chess. The more imaginative player will produce more beautiful, or artistic, games at the chessboard than the player who has a totally scientific approach to the game. But, the more imaginative player will also suffer some crushing defeats. I back up this statement by comparing two great chess Grandmasters who were contemporaries: Ludek Pachmann (1924-2003) and David Bronstein (1924-2006). Pachmann was the greatest chess openings theorist of his time, with an encyclopaedic knowledge of the openings. Bronstein was more of an intuitive player, constantly looking for new twists and sharp ideas in the openings, and in the middle-game and end-game too. As players, Pachmann and Bronstein had very different results over the chessboard. Pachmann was very difficult to beat, but the majority of his games ended in draws! Bronstein would risk defeat by taking chances, and often produced brilliant and almost logic defying victories, against the best players in the world. Pachmann was a world-class Grandmaster, but never came close to becoming World Champion. Bronstein's exciting play won him many admirers, and saw him tie a World Championship match 12 points each, against the great Mikhail Botvinnik in 1951. Botvinnik kept his title by the skin of his teeth! So, by challenging accepted knowledge within chess theory, Bronstein reached the greatest of heights (even though the world title eluded him), while Pachmann achieved great status as a theoretician, but his dogmatic approach limited his achievements over the chessboard. By not taking risks, Pachmann had to endure a great many drawn games, rather than dramatic wins. In conclusion, my point is that we often tend to accept known theory without question, when there may indeed be several questions we could still ask. The great mathematician, Copernicus, stated that the Sun was at the centre of the universe, and produced mathematical evidence that supported his theory. We now know that Copernicus was wrong in this particular case, although theory seemed to back up his claims, until he was proven wrong! I'm sure that certain aspects of mathematics could still be challenged in the future.

[Nolo Segundo]

You write so well of complicated things, Rick, but 'Man is just a machine' ?-- ah, if it were only that simple.

To borrow from the Bard some more:
'what fools these mortals be,
expecting to find God hiding
behind some tree...'

[Tomás Ó C..]

Wow... most of that is right over my head and Im going to have to read through it a few times to grasp but a whiff of it...

I do hope you put some of these in book form even if only for posterity - and get the PDF up on GoogleBooks so its not lost should this site go wallop like it did in 2008...

My head is wrecked myself looking up the Irish and our local connection to the slave trade in the carribean - not just as slaves but as slavers, alas this is true...

I think I've some up about it on the site - if not its on my website www.writingisnrhyme.com

[Tate Morgan]

The more I think on the imposed question of whether a man can think too much for his own good the more I feel I sadly prove my own point I hate to lose an argument with myself yet it seems to happen more often these days than in my youth when I was always right

[Rick Puetter]

Hi Tate,

As you know, I generally don't publicly comment on reviews, but prefer to send the reviewer a private e-mail to express my thanks and thoughts. However, if I feel that my response carries on the theme of the piece, and might be of general interest, I open the response publicly as I am doing here.

Yes, the discussion of whether mathematics is a "universal language" and bears truths beyond human endeavors is in complete keeping with the ideas expressed in my essay, and is at the heart of the matter. When I was younger, I firmly felt this was absolutely the case. It was what drove my love of mathematics, and it was what caused me to enter college as a math major. But as I also had a great love of physics, and because all of my close college friends were physics majors, I eventually switched my major to physics as well. What a fortunate turn of events this was, as my real interests truly lied along the lines of physics. Still, my love of mathematics continued forward, but my need for understanding reality slowly was driving a wedge between me and mathematics. I didn't understand it at the time, and only now, some 40+ years later, am I starting to really understand why.

There are lots of conceptual problems with mathematics that disturb me greatly. At the heart of it is an Aristotelian view that all of the world's truths (in this case mathematical truths) can be derived from purely thinking. Such a view has failed miserably for physics, and we have learned that physical truth can only be won by doing countless experiments, and that further our intuition and pure thinking tends to lead us astray. We do not understand this world. This world is beyond our intuition, and as Heisenberg famously warmed: "We must stop trying to understand it". Simply put, our thoughts fool us into believing we know how things should be, and I am now wondering if mathematics has fallen prey to just this fallacy.

Now what do I mean by this? Let's take the integers, for example. Integers were developed by early man for counting. He wanted to know how many apples he had or how many sacks of grain. From this we developed simple arithmetic: 2 apples plus 2 apples equal 4 apples. Very simple stuff. But we never intended this to be extended to infinity since no one ever has an infinite number of apples. So does this concept (counting) even apply to infinity, let alone uncountable infinities? And remember, even the simplest of notions have profound effects on the properties of mathematics. For example numbers also are used the designate order, i.e., this one is first, this one is second, etc. Counting numbers (how many) are called cardinal numbers. Numbers that designate order are ordinal numbers. They are not the same thing and this difference becomes surprisingly important when we get to infinities as ordinal numbers have lots of additional properties relative to cardinal numbers--a very surprising and perplexing fact.

So it is easy to introduce things that seem innocent, but which have tremendous impact if you follow through on all the consequences. The more complex the subject, the more profound the consequences and perhaps the more questionable the results. Perhaps there needs to be more: "Oops, I didn't mean to imply that" moments in mathematics, where we back up instead of pushing forward with all possible speed.

What do I mean by this? Here is an example (there are many). Group theory one of the crowning achievements of mathematics, and one of the simplest systems to study, has such issues in my view. I think here we get into stuff that surprises us and should give us pause to perhaps say: "Oops, I didn't know this would happen. Let me rethink that."

Today, people claim these days that finite, simple groups have been completely classified (see the immensely interesting article in Scientific American of a few months ago, "The Whole Universe Catalog", July 2015). The article describes four aging mathematicians, group theorists in this case, that are rushing to write down the theorem that classifies all such groups, as these are the only four people in the world that have complete understanding of the theorem that spans an estimated 15,000 pages. Wow! What an achievement. And many of the 4 classes of finite, simple groups that arise from this work have very powerful applications in solid state physics, particle physics, engineering, metallurgy, etc. This mathematics brings order and understanding to a diverse set of physical systems. But lurking in the corner, sitting out of view, hiding perhaps, is the set of "Sporadic groups". This is not a family of groups with an infinite number of members like the Lie group, for example, that describes all the geometric symmetries in the world, but the sporadic group has only 26 members. And what weirdos they are! Let's concentrate briefly on the "Monster group"--yes, that's what they call it. It is the largest of the sporadic group members, which all have a finite number of elements. The Fischer–Griess Monster has 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 ~ 10^54 elements. Why you ask? Maybe because God wanted it that way. Does it have any meaning or purpose? God only knows, but physicists are also starting to write papers about it. These papers are esoteric for sure, but does this have any real meaning? Is this one of those "Oops, I didn't know that this would happen" moments? Now the mathematicians almost all believe in the reality of this Monster. Again, I'm not so sure. I'm wondering if this arises out of taking things too far, as I would claim with infinities, which were never intended when we were back in the orchard counting apples. So what is it? Well, I simply don't know. But I don't believe in Shakespeare when he has Hamlet say to his friend Horatio: "There are more things in Heaven and Earth, Horatio, than are dreamt of in your philosophy", Rather, I think it is the other way around, i.e., Man can dream up many more things than are real.

Just some additional thoughts, Tate. Thanks for your review.

Best regards,

Rick

[Tate Morgan]

I am still soaking that in one thing I get for sure out of this little. Talk matter how complicated intricate and divine this universe is there is always an answer and a reason we just haven't discovered as of yet so why do you think mathematicians are so more likely to believe in God could it be because their line of thinking has to have order and the universe seems a chaos to them a physicist has the added dimension of seeing the beauty in this universe and appreciating simply that it exists mathematicians can't stand to not know the answer so when they realize their lives are too short to figure out the answers their vanity grants that grace to God whereas a physicist can take credit for every new discovery and leave the mundane of why to the mathematician

[Rick Puetter]

I think the reason mathematicians are more religious than physicists is pretty simple. Mathematicians have a unshakable faith in the truth of mathematics, a truth which is beyond Man. And this leads pretty easily to other unshakeable beliefs, like a belief in God. Now remember, mathematicians are not scientists. They don't question their world. Physicists take the very opposite point of view. They question everything. They want to know: "Well how do we know this? What evidence do we have?" And even after they have a positive answer, they try to peel it back at the edges to see if what they think they know continues to hold up. So to physicists, having an unshakable faith means you are a bad physicist and you better get out of the profession. So a physicist would want to know why you believe in God. And not having an alternative explanation is not a reason. A physicist would choose to simply say: "Well, I guess I don't know if there is a God", but believing in God is not an acceptable alternative. Now a physicist would entertain arguments about plausibility, for example they would entertain thought experiments about the existence of alternative, parallel Universes, even if we don't currently have evidence of their existence. However, such ideas would have to offer some benefit in understanding things and would have to provide a mechanism for this understanding. The idea of parallel Universes does offer such explanations, a powerful one is an answer to the anthropomorphic question: "Why does our Universe have the properties that it has and why do we exist?" Belief in God also provides an answer to that question, but it doesn't offer a mechanism that we can understand. Instead, religion offers us mysteries and says that God's ways are beyond our understanding. Very well. If they are beyond our understanding, then a physicist would never believe in them. Why would he/she? In that case any other alternative explanation, e.g., the principle of gobbledygook, is just as valid an explanation since we don't understand either one. And even when we have a proposal that offers a mechanism, such as the multiverse, this doesn't become science until it is tested. The multiverse is far beyond that right now. Even though we think we might understand how that might arise from string theory, we won't believe in its voracity until we have measurable consequences that were predicted by the theory. This is very different than religion and it's different than mathematics as neither require physical proof to go forward.

[Tate Morgan]

I would go so far as to
Postulate that she is the source of your elusive dark matter lol

[Tate Morgan]

While we may be a sort of machine I have to think that mathematics is the universal language while we are an enigma case and point you and I would make great debators as we are like old dogs with a bone when arguing a point My ex wife in the other hand thinks emotionally not logically she is ruled by that unkind spirit no matter my evidence to the contrary once she has made up her mind logic facts mean nothing when posed against the impenetrable declaration that no matter what I say she doesn't feel that way hence it isn't true. Now you can imagine the frustration of a prosecuting lawyer trying a case with her as a juror to her emotion feelings based upon her mood are fact logic is fiction hence I had to divorce her for my sanity lol in the past discussions you and I have had about the validity and possible makeup of black holes I would just like to say I fare much better in those discussions with you even though you so outrank me in education than ever I did in an argument against that impenetrable dead end that I once called my wife lol

[mattavelli]

Hello, Rick. I enjoyed your article. I didn't follow any links, so I feel like I missed a lot; but you presented some fun riddles to ponder.

"Simply put, Gödel demonstrated that arithmetic was incomplete, and that there were mathematical statements in arithmetic than((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?"
I think the answer is, it's an opinion. Haha
This reminds me of a group assignment from intro to philosophy. The topic was determinism vs. indeterminism. We were to prove either. I recall thinking that neither position affected or was a condition for anything real. And with that idea stuck in my head; I slacked off.

“Is man just a glorified machine, or is he something more"?
Man wants. I've never seen a machine do that.

Thanks for sharing.

[Great Aunt Astri]

You are a scientist Rick. I would love you to put some of the science and maths issues into a poem (with poetic licence, of course!) so that all of us could have an inkling of understanding. I once wrote a poem after my daughter, doing her homework, told me that something or other (was it a positron?) had a ghost partner. My poem was called 'It is time for us to part company' Unfortunately I wrote it on my old computer and did not make a copy.

[Rick Puetter]

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

[Great Aunt Astri]

Thank you Rick. I will look up all the references, for although I am an amateur I am really fascinated by all this. There is one book you may be interested in. Stupidly I have forgotten the name of the author. He was British and had a background in mechanical engineering, but he wrote am amazing book (the name slips my mind) all about alternate views to the quantum theory and modern ideas. He wrote about everything have a 'frame' and about 'folds' in existence. It is a really enlightening book and I will rack my brain and send you the Authors name and that of his book. He is contemporary, probably about your age.