Friday, September 2, 2011

On Left/Right Symmetry, the Platonic Ideal, and Godel's Theorem

[Another break from engineering today...more on symmetry...and asymmetry]

Plato, who lived in Athens from (429 to 347 BC), believed that there exists a realm of perfect, ideal shapes. He believed that the shapes we see, such as the circles we try to draw on pieces of paper, are but imperfect Earthly forms of a perfect, eternal ideal that we can't see.
The idea that everything is but an imperfect example of a smaller set of pure objects might be fine and dandy for certain mathematical objects, such as the circle or the five Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron), but the idea of a perfect object breaks down for living creatures. Unfortunately, the philosophy of idealism as developed by Plato and Socrates hindered the eventual development of the theory of evolution because many people believed that each species was related only to its eternal ideal form, and not to other species. I will point out in this post the error in thinking that there could be such a thing as a perfect cat or a perfect elephant or a perfect homo sapiens.

Throughout the history of life, there have been "either/or" symmetry-breaking moments. One of the first examples was when life forms used and grew only left-handed amino acids, instead of right-handed amino acids. (For more information on the difference between left and right handed molecules follow this link.) I think it's important to stop and ask: why did life develop the ability to use only left-handed amino-acids? (Note that there are a few bacteria that use right handed amino-acids, but the overwhelming majority use left-handed amino-acids.) It turns out that this is still an open question because the evidence scientists have collected to answer the question doesn't directly confirm any one theory.

Here's the main evidence collected so far to try to answer this question.
1) The Miller-Urey experiments produced both left-handed and right-handed amino acids in equal amounts. This suggests that the left vs. right choice was a fluke, i.e. there was a 50%-50% chance that one type would dominate, and perhaps there was a 50% chance that we would have used only right-handed molecules instead.
2) Meteors seem to produce 7%-9% more left-handed than right-handed amino-acids. This suggests that perhaps left-handed amino-acids survive better than right-handed one, but even a 7%-9% initial difference means that there's still a large chance that life could have developed the use of right-handed molecules instead. Even so, we are left with multiple questions: a) why aren't ~40% of the bacteria able to right handed amino acids? b) why can't bacteria use both left and right handed amino acids...and both left and right handed sugars? c) why are there more left-handed than right handed amino acids on the meteors? d) What is the initial cause of the asymmetry and is the cause of the asymmetry in the meteors the same as the asymmetry on Earth.
3) We know of only one force in nature than can distinguish between left and right, and that's the weak nuclear force. This force does not conserve parity, i.e. the weak nuclear force can be used to distinguish left and right handed particles. For example, only left-handed neutrinos or right-handed anti-neutrinos can participate in reactions via the weak nuclear force. No other force is that unique (not gravity, not electromagnetism, and not the strong nuclear force.) But what does the weak nuclear force have to do with anything here on Earth? Or on the meteors mentioned above? Does the weak nuclear force have anything to do with why life developed the capability to use a certain chiral form of amino-acids, DNA and sugars, but not the opposite chiral form? I can't think of how this would be possible because life does not appear to be dependent on the weak nuclear force...only gravity to hold us on Earth, E&M to hold protons and electrons together, and the strong nuclear force to hold the nucleus together. I can't think of any way in which the weak nuclear force could effect the structure of large molecules like DNA, amino acids and sugars. (But if the weak nuclear force doesn't effect life, how does a human zygote nearly always develop such that the heart of the fetus is on the left side of the body? How does the zygote distinguish between left and right if gravity, E&M and the strong nuclear force can't distinguish between left and right? Does it rely on the handedness of DNA and amino acids?)

So, this leaves me with even more questions: Was the eventual dominance of left-versus-right due to initial offsets? (Perhaps the universe itself is asymmetric) Or more likely, was it just dumb luck? And by dumb luck, I mean: was it the particular chirality (i.e the handedness) of the first RNA or DNA that caused the exponential growth of a certain type of amino-acid via self-replication.

But there were so many left-right choices in the history of human development, that I'm pretty much forced to conclude that it was dumb luck, i.e. we could have developed the use of right handed amino acids. Other left-right 'choices' include: a) which side our heart develops b) which side of our brain develops certain storage locations  c) why 90% of the population is right handed, and  d) why the left testicle for men hangs lower than the right on average

The point I'm trying to make here is that there have been so many left-right (or up-down) choices in the history of life that there's no way to think that the left-left-up-right-down-right-up (etc...) that makes current humans was due to anything but dumb luck.

But how does this seemingly breaking of left-right symmetry stand up when compared to the Rosen-Curie Principle, which states that the symmetry of the effect is greater than the symmetry of the cause. If life can only become more symmetric, then how can we account for the dominance of left-handed amino-acids? Why does only one type of amino acid grow exponentially? In order to not violate the Rosen-Curie Principle, we must accept that the universe was not perfectly mirror symmetric to start, because if it were perfectly mirror symmetric to start, then there would be no way to have only left-handed amino acids grow exponentially. This suggests that the initial conditions of the universe were not left-right symmetric. It is very likely that the universe started in a state of low symmetry.

And now, I'd like to show how Gödel's incompleteness theorem effectively destroys Plato's idea that there could be such a thing as an ideal or universal 'tree' or 'bird.'
Gödel showed that, for sufficiently strong and well-formed set of axioms like the set of axioms for arithmetic, there are statements (that we'll call 'G') that can't be prove to be true or false. And further, if we now add a new axiom to the set of axioms which states "G is false" or an axiom which states "G is true", there will be new statements (that we'll call F & H) that can't be proved to be true or false in the new sets of axioms. We could a set of axioms with "G is true" and "H is true" there will be a statement 'J' which can not be proven to be true or false, and this means that we can now add either "J is true" or "J is false" and make a new set of axioms...which will still be incomplete...i.e. there will still be statements out there which can't be proven to me true or false.

And now, I'd like to make an analogy with life.
It's as if life is a program for generating new programs. When it reaches an axiom (i.e. food) that it can't determine whether it fits into its existing program, it has to make a choice and either include the axiom as true (i.e. left-handed food) or false (i.e. right-handed food.) Once it makes the decision, then all future programs make statements and new programs assuming that the axiom is true (or false.) There's generally no going back. Then, later on, it runs into a new axiom. Will it accept the axiom as true (create left-hand amino-acids) or false (create right-handed amino-acids)?  There's no right answer to the question: should we accept this axiom as true or false? It's like we're little Gödel-esque programs constantly trying to calculate the truth or falseness of certain statements, and when we come upon a statement G that we can't prove is either true or false, we simple make a choice and incorporate either "G is true" or "G is false" into our set of programs.
The analogy here is between "truth & false" and "right and left handed." (This is not intended to be a perfect analogy; it's only intended to convey the sense that there seem to be a lot of times in the history of life in which there were completely arbitrary choices made between left-handed and right-handed chemical, just as the choice of either adding the statement "G is true" or adding the statement "G is false" is completely arbitrary. There was no way to prove that G was true or false, so you just pick one and move on.)

In the case of mathematical sets of axioms, you have to pick "G is true" or "G is false." You can't pick both. (Or else you don't have a consistent set of axioms.) But is the same true for life? Could bacteria have developed the capability to harness both left and right handed food molecules? Perhaps it did initially. But if so, why did that ambidextrous form of life not out grow and out multiple the only-left or only-right handed lifeforms? You'd think that a bacteria that could use both left and right handed molecules could grow faster than a bacteria that could only use one handedness of molecules. What's the advantage to just-left or just-right molecules?

My guess is that trying to use both types ends up creating DNA molecules that are both right and left handed, and since left-handed DNA will not match up with right handed DNA, there might be a problem with trying to use both right and left handed amino-acids. The use of certain sugars probably also stems from the fact that sugars are the backbone of DNA. You can't afford to have waste your time building a strand of DNA and then have a wrong-handed sugar molecule show up and ruin the spiral.

Life 'choses' one handedness over the other, and this happened over and over and over again throughout the development of life.

And so now we see the problem with Plato's idea that there is a world of ideal forms. There is no such thing as an ideal form for living creatures because an ideal form can't be arbitrary. If life is made up of sets of arbitrary choices between left and right, then there is no such thing as an ideal life form because how could an 'ideal life form' be arbitrary. For example, would the ideal 'tree' use left or right handed amino-acids? Would that same 'ideal tree' spiral clockwise or counterclockwise as it grew?

Each living creature is like those sets of axioms that Gödel thought is made up a series of completely arbitrary choices between left and right. Living creatures are irreducibly complex. There are just too many arbitrary left-right choices, and each of those left-right choices means that there is no ideal species.  For example, would the ideal human have their heart on the left or the right? Would it be in the middle?

As the physicist Roger Penrose wrote, "If the axiom of choice [i.e. "G is true" or "G is false"] is a mere matter of opinion or of arbitrary decision, then the Platonic world of absolute mathematical forms contains neither the axiom of choice nor its negation." Penrose is suggesting here that mathematics (such as algebra) can't exist in some Platonic ideal world becuase the choice of axioms are completely arbitrary. We can continue with Penrose's thinking in "The Road to Reality"... There is no such thing as a generally true mathematic statement. There's only statements that are true if you assume axiom x,y,z,...  are true. But if you assume a different set of starting axoims, then the statement might not be true.  For example, is the statement "The sum of all angle for a triangle equal to 180 degrees" a true statement?   It's true if you assume certain axioms (such as for planar geometry), but it's not a true statement if you assume different axioms (such a hyperbolic geometry.) So,we can't even say that "The sum of all angle for a triangle equal to 180 degrees" is always a true statement.
Is there somebody to tell us which set of axioms is true?   The answer is no. Can God tell us whether the statements of mathematics are always true?  The answer, once again, is no.

The laws of mathematics are not handed down to us from God or some Platonic realm of ideal equations. And further, Plato's ideal world can not hold life forms because it can't contain such impure 'left-handed' creatures, such as we are. While it's tough to ponder the abundance of arbitrary choices between left and right, I think that it's an interest topic to discuss between there are a lot of open question:
1) Why did life develop the capability to use only left handed amino acids? (Nearly only left)
2) How does a zygote distinguish between left from right if the three forces that effect life (gravity, E&M and SN) are left-right symmetric? Does the zygote use the chirality of molecules to tell the difference? In which case, we go back to question #1.

What's so ironic is that Gödel considered himself to be a Platonist, i.e. he believed to some extent that mathematical truths exist outside of this world. But his first and second incompleteness theorems, as well as his work on arbitrariness of the axioms of set theory, are the last nail in the coffin of Platonism because his research proves the arbitrariness of the axioms that we use in mathematics, and this is very similar to the arbitrariness of many of the left-right 'choices' that life has made throughout its exist on Earth.

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