THE LIMITS OF MATHEMATICS

“…mathematics…ought only to give definiteness to natural philosophy, not to generate or give it birth.”[1] 
Francis Bacon

“So far as the laws of Mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality.” 
Albert Einstein, Geometry and Experience

 While the postmodernist social scientist often uses the language of words to obfuscate, the postmodern 'hard sciences’ scientist sometimes appears to use the language of mathematics to obfuscate. Mathematics is a language not a science. It is the language of science and of the known physical world. The inorganic reality of our known world can be described mathematically with eerie precision. This we know is an absolute fact. I stress ‘known world’ because we cannot know empirically that mathematics pertains for all of nature. To know this empirically, one would have to be outside of nature, to be a supernatural being, to be a supernatural God.

 Yet while it has been a fundamental tool in researching the inorganic physical world mathematics has been unable to access or discuss life as such mathematically. We can mathematically describe most of the commonplace aspects of life (blood pressure, blood sugar, cholesterol, etc.), but not life itself. What makes something alive, rather than dead or not alive is still beyond our comprehension. We know that life has evolved (evolution is an absolute fact), and we know that neo-Darwinism has been the mainstream working theory of evolutionary scientists, producing ever increasing (but still incomplete) fitting explanatory results[2]. Even so, a small but growing number of scientists, foremost amongst them Lynn Margolis, are questioning the neo-Darwinist model. We know mathematically and empirically what happens as life becomes more complex. We know about genes, chromosomes, reproduction, population, but life qua life is still the greatest mystery within the greatest mystery of existence.

Mathematics is the abstraction of reality; it is not reality as such. As author Howard Bloom says: “…abstractions may be indispensible but they don’t accurately reflect reality”. [3] Theoretical cosmology and theoretical physics, being essentially mathematical constructs, are besot with abstractions – indeed, they are one big abstraction; they help us relate to reality, comprehend it, manipulate it, but they are not identical with reality.  They are descriptions of reality (and sometimes of non-reality). Theoretical mathematics can also deal with non-reality (re: Escher Bach) as efficiently as it can with the phenomena of reality, but not necessarily with reality as such.

 In other words Reality can be described mathematically but that description is not reality as such.  Mathematics is the language, the grammar of reality and since math can rationally describe a non-reality just as well as reality, non-reality is often mistaken for reality; something that cannot happen in the everyday language of human intercourse.

 For example “The Earth is flat” is a perfectly grammatical sentence; it reflects the logic of grammar, it is internally consistent. But just because we can say it grammatically does not mean that the earth is flat. As Kenneth Boulding put it “Mere internal consistency is not enough, for there may be views of the world which are internally consistent, but which are, nevertheless, not true, in the sense that the real world does not conform to them”.[4]  This confusion between reality as such and mathematics as the language of reality has become a real stumbling block to the advance of human knowledge as we close in on the border between physics and metaphysics; the border between the empirically accessible finite and the inaccessible infinite.

Mathematics is especially obstructionist if we assume evolution to be the grand narrative of ultimate reality (not only the descriptive metaphor for the development of organic reality).  Evolution is characterized by qualitative change, by the dialectic of the internal unfolding and development of reality.  Mathematics can only deal with quantitative change and movement, with the externals of ultimate reality as we perceive them, with the phenomenal world not with the noumenal world, with the appearances of the world not with the essence of the world.  This is the essential message of Kant’s Critique of Pure Reason. Kant uses reason or rational (from ratio) in its precise technical sense, referring to the laws of non-contradiction inherent to mathematical ways of thinking. There is a limit to knowledge not available to empirical investigation but only capable of mathematical description (pure reason).  With this critique he sets the stage for Hegel’s dialectical way of thinking; a mindset better suited to evolutionary development than Aristotelian stasis.

 Reality is much richer than the mathematical language we use to describe it.  Indeed mathematical language tends to desiccate the luxuriant richness of reality. A good prosaic example of this is economic theory (which is as mathematical as quantum physics) as it relates to the reality of the business world and human economic activity. Business is not just bookkeeping or a reflection of economic theory; it is passion, belief (George Bernhard Shaw saw it as a religion), imagination, falling in love with your own creativity; it is often a sense of mission reflecting a sincere desire to change the world for the better.

 Starting a business is an act of creation no less than an artist painting a mural or a musician writing a symphony. This is because human creativity is indivisible; whether used to make a better mousetrap, plant a different crop, paint a picture or start a new business it stems from the same source: the human desire, unique amongst species, to make sense of itself and its environment and leave its mark on the future.  Human creativity is its own justification and needs no other sanction; it is what makes human beings human and thus godlike. Gods and humans create.  The Old Testament God created cosmos (order) from chaos and human beings create cosmos (paintings, music, statues, products, manufacturing systems and services) from chaos. In point of fact the Old Testament God did not create us in his image, we created him in our image – ‘God’ is an induction, not a deduction.

 Mathematics laden economic theory is useful and even essential for the formulation of economic policy but it is utterly insufficient for understanding what is really happening in society and the economy. Empirical proof of this is that economists rarely if ever succeed in the real economy of products and services.  At most they can serve a useful function in the financial sector. I cannot think of one economist who ever started and ran a new business, let alone one who created a whole new sector of economic and business activity.  Economists are limited by the ‘rationality’ of their mathematical way of thinking. Since one cannot mathematize imagination or creativity or passion (the hallmarks of the entrepreneur), or comparative social energy and cultural values, they are stuck in their analytical thinking and prevented from entering the world of synthetic thinking intrinsic to the artist, inventor and entrepreneur. And yet it is the inventors and entrepreneurs who create economies, not economists.

 Another instructive historical example of the limitations of mathematics would be the mathematical models McNamara and the Pentagon used to prosecute the war in Vietnam and the catastrophic failure this led to.  These models had nothing to do with the human passion on the ground. McNamara could never mathematize the Vietcong and North Vietnamese refusal to lose and the cultural differences that explained their ‘willingness’ to suffer loses inconceivable to the American mind.

 This is one reason why the hyper-rationality of intellectuals is suspect in the eyes of the ordinary human being – too much abstraction, not enough passion and gut feeling. Non-intellectuals care little for learned sociological, psychological and anthropological explanations regarding the source of their patriotism and veneration and why they love their children more than others and biological explanations of why they fall in love.  They deeply resent the tone of superior understanding implicit in these explanations and are deeply suspicious of the analytical coldness of the explainers – they sense there is something non-human about it.

 This explains the non-rational need for literature and poetry and love. Rationality is never more productive than when it sails on the ocean of the non-rational imagination. The daydreaming of Newton and Descartes and others has been the necessary fertilizer for scientific flowering. The unsophisticated imagination of Edison and Ford and others has been the necessary fertilizer for economic flowering. Bronowski in the television production of The Ascent of Man cited the three ‘Bs’ as the source of creativity: the bedroom, the bathroom and the bus. Einstein was hard put to explain the source of his own revolutionary ideas; his explanation was reminiscent of one of Red Skleton’s characters (Klem Kiddlehopper) who said: “I was just standin on the corner doin nothin and it hit me”

 Reason is not the source of creativity; reason is used to explain non-rational intuitions in a rational way. It is not the cause of ideas; it is the instrument necessary to turn ideas into effective tools. It is a necessary tool but certainly not one sufficient to explain the human condition within ultimate reality.

 The Tyranny of the Law of Parsimony

“Frustra fit per plura quod potest fieri per pauciora [It is futile to do with more things that which can be done with fewer]”
William of Ockham

 “If a thing can be done adequately by means of one, it is superfluous to do it by means of several, for we observe that nature does not employ two instruments where one suffices”  
Thomas Aquinas

but

“Seek simplicity, and distrust it”
Alfred North Whitehead

If science were to have a ‘religious’ dogma it would have to be the Law of Parsimony (LOP) otherwise known as Ockham’s (or Occam’s) Razor which popularly (but inadequately) stated means “all things being equal the simplest explanation is usually the correct one”. I italicize ‘usually’ because this caveat is usually forgotten or ignored.  This is so because of the tremendous success of the LOP in advancing human knowledge and by this advancing human welfare. It has been (and still is) the standard by which physics, and its language mathematics, consistently judges itself; the search for reductionist simplicity, elegance in explanation and theory being the standard by which it judges itself.

There are many fields of current inquiry in which the principle of simple elegance is unfortunately ignored. If sociologists were to follow its strictures they could hardly write at all (which might be a blessing). And while opposition to the LOP has been constant since Occam evoked it[5], it still seems to dominate fields of inquiry that presume to be scientific.

Ockham and Aquinas might have been accurate if they were referring only to physics. The simplicity of the Copernican hypothesis certainly destroyed the ever growing complexity of the Ptolemaic universe and laid the groundwork for modern cosmology. Einstein’s E=MC2 is the epitome of parsimony erasing the complexities of ether theory – although later Einstein was flummoxed by the complexifications of quantum theory. But the LOP is self-evidently insufficient when it comes to issues related to biology/evolution as well as to anything related to historiosophy and psychology (Marxism and behaviorism being the biggest violators: oversimplifying complex phenomena to the point of absurdity).

Nature/evolution is constantly generating endless “solutions” to the challenges of our hostile environment, even when prior solutions worked (and often still work) quite well (sometimes even better). Evolution is an ongoing dialectic between the ‘inorganic’ world molding the ‘organic’ world and the ‘organic’ world changing the ‘inorganic’ world which then opens new niches for the organic world. Biological evolution is not linear, proceeding in lock step to some ‘final cause’; it is not always creating ever more advanced species or advanced versions of species to fill certain niches. It will sometimes produce species more primitive than ones already extant in order to ‘test’ if it can fill a certain niche more efficiently. At the macro level evolution tends to ever greater complexity and sentient advancement; the ongoing complexification of existence over cosmic time seems to be self evident, but the “daily” business of evolution does not concern itself with its greater cosmic ‘mission’ – it busies itself with experimenting with endless varieties of life to ‘test’ if they will work.

 The vocation of evolution is complexification; the avocation is endless tinkering, trying out various forms of life that will fit into the endless niches of nature. Chaos theory will give us some insight here – environmental fractals constantly create new niches for new variations of life. The very existence of these fractals function as a local final cause drawing/driving evolution to develop new life forms and new versions of similar life forms capable of filling them.  These life forms and their various versions may or may not be more complex than previous versions (local evolution is indifferent to the general trend towards complexification of cosmic evolution).

 Darwinism is to evolution what Copernicus was to cosmology and neo-Darwinism is to evolution what Galileo was to cosmology.  Unfortunately, in battling the nonsense of the creationists many evolutionary biologists have made neo-Darwinism an absolute dogma ignoring opposition to it within the scientific community (see Lynn Margulies).

 This brute fact of evolutionary history renders Occam’s Razor suspect.  If there was one elemental cause of natural (and social) phenomena Occam’s Razor might have been germane.  But since we observe a multiplicity of solutions (each with its relative advantages) then we must conclude that natural (and social) phenomena often have multiple causes.

 In regards to the social “sciences” – those intellectual endeavors that deal with human history and society – academes’ simplistic, reductionist explanations/theories that explain everything (economic, social, political history) are so intellectually impoverished as to be absurd. The instinctive anti-intellectualism of the worldly layman when confronted with these theories is, as William James noted, quite healthy.

 The various schools of history are a case in point.  What were the causes of America’s Revolutionary and Civil Wars? Any real reading of the facts and events reveals that they were a combination of political ideals, economic interests, cultural differences, accumulated resentments, personal ambitions and divergent social/ political/cultural histories.

 Regarding evolution and human society I would turn Occam on his head and claim that “all things being equal the most complex explanation is most likely to be closer to the truth”. Complexity can contain and entertain more nuance and subtlety, more ambiguity and paradox than a parsimonious explanation. Life per se is nothing if not nuanced and subtle; human society is nothing if not ambiguous and paradoxical. If evolution is true it must perforce contradict the law of parsimony. I am not talking about a hesitant almost apologetic allowance/admission that some explanations do not conform to the Law of Parsimony.  Rather I am advocating the full fledged enthusiastic support for the most complex explanation.  I realize that the present and near future generation of scientists will have trouble with this notion because of the inertia of the sociology of science.

 But if Nature’s “existence” is infinite in all direction and for all time, as Dyson claimed, then its complexity (as a whole and in all its parts) is also infinite.  The LOP only pertains to very small arbitrarily determined space/time domains. Within this restriction it has been extremely effective and has advanced civilization more than any other principle in human history.

 But as we approach a largeness that is the “outer” boundary of our Cosmos and a smallness that is the “inner” boundary of our Cosmos the relevance of the LOP as a guiding principle to our endeavors begins to breakdown. As with mathematics it begins to be a “neo-scholastic” barrier to further understanding.  Quantum physics has already revealed the ultimate complexity, ambiguity and paradoxicalness of inorganic reality (and might eventually have something to say regarding evolution). As with reductionism the LOP still has a valuable role to play in furthering our knowledge of clearly defined domains, but as we approach the border between physics and metaphysics its methodological validity begins to dissipate.

 The LOP was necessary in the history of thought to clear away the rubble of superstitious mumbo jumbo – it is still a valuable tool in detecting and clearing away the rubble of superstitious mumbo jumbo of much of academic writing in general and that of the postmodernist sort in particular. But once it accomplished it historic task it left a smooth surface – “elegant” in its reductionist simplicity but meaningless to any effort to describe the complexity of existence.  It was so reductionist it reduced living reality to nothing. It became a formal system as in mathematics – internally consistent but irrelevant for the “I/Thou” relationship that individual human beings have with the infinite complexity of the brute fact of their own existence vis-à-vis the brute fact of existence itself.

 The new science of complexity has come to help us to build a magnificent structure on this smooth surface (prepared for us by the LOP) and dig a deep foundation below. We are prepared to grow out of our intellectual infancy and advance into our cosmic majority.

 Conclusion

 Has the LOP run its course?  Has it exhausted its great historic task in the cause of human progress? Is it now a great barrier preventing significant breakthroughs in human understanding? Is there a historical analogy to Scholasticism in that the great medieval synthesis enabled tremendous intellectual progress up to a point (even producing the LOP) and then became a barrier to further intellectual progress – from which the West was only rescued by the Renaissance?

 The LOP applied to historical causation has been a major factor in the poor intellectual integrity of the social sciences and even to some degree in biology and evolutionary thinking of every degree. A Marxist historian will twist all facts to fit into Marxian theories of history and economy – the explanations will be internally logical and very difficult to refute but this will be because they will be one dimensional to a very great degree. Neo-Darwinists and Big Bang cosmologists will often do the same. As Sidney Ratner wrote “…even professional scientists and philosophers become so absorbed in defending some vested interest or eloquent idea that they do not welcome truth, but seek victory and the dispelling of their own doubts”.[6]

Notes#

• The roots of LOP are deep – e.g. Thales “everything is water”
• The truth is that NOTHING is only one thing or even a multiplicity of things. Even reductionism in Physics only enables enough information to manipulate nature (it never enables one to reach “the thing in itself” – as Kant pointed out)
• I would allow the LOP a monopoly in only one area – mathematics. Yet even here I expect some philosophers of mathematics might disagree with me – e.g. Gödel’s theorem.
• The LOP is not good for any endeavor that cannot be mathematized.
• If you can mathematized something it pertains – life, biology, evolution, human history, political science, climate cannot be mathematized – therefore LOP as applied to the social and evolutionary sciences is an obstacle not an aid.
• The place of consciousness. Behaviorism and the reduction of human consciousness to chemical interactions are absurd enough to self-evidently prove the impoverishment of LOP
• Is the LOP one of the foundations for totalitarianism – ONE logical way to do something – no bourgeoisie messiness (Constitutionalism is sloppy and imprecise) – just a simple explanation. Is the LOP inherent in every dogmatism?
• The ecological model of postmodern civilization must liberate itself from LOP if it is to optimally fulfill its promise. Indeed the LOP runs counter to current ecological wisdom and might be one of the sub-conscious causes of mono-culture farming that has caused so much ecological devastation worldwide.
• The plentitude of “The Great Chain of Being” is not the opposite of parsimony – complicated is.
• LOP facilitates the tendency to cherry pick facts that re-enforce our own theories. It is counterproductive to the radical skepticism that must always be the foundation of all good science.
• The risk of rejecting the LOP is that we might fall into the trap of post-modern blah blah!
• Cancer is caused by complexity not reductionist simplicity, by a countless combination of causes not by a reductionist one cause.

[1] Novum Organum xcvi

[2] Stem cell research is now revealing that epigenetics changes DNA and that these changes can be passed on to next generations.  The inheritability of acquired characteristics has been resurrected and Neo-Lamarckianism is now a complementary to Neo-Darwinism.

[3] Bloom, Howard. The God Problem; Prometheus Press, Amherst New York, 2012 Pg 28

[4] Boulding, Kenneth E. “Towards an Evolutionary Theology” in Science and Creationism Pg 146

[5] http://en.wikipedia.org/wiki/Occam’s_razor

[6]  Ratner , Sidney. “Evolution and the Scientific Spirit in America” in Science and Creationism (ed. Ashley Montagu) Oxford University Press 1984 Pg 411.