Recently, I listened to a series of recorded lectures by Steven L. Goldman titled Science in the Twentieth Century: A Social-Intellectual Survey. In these lectures Goldman, who has degrees in both physics and philosophy, makes some interesting observations about the development of science and math. In the present essay I will take some quotes from his lecture notes and comment on them. First of all, he makes the general statement that
"The practice of science was traditionally open and democratic, but by the late 1800s, physical science had taken on the characteristics of an esoteric cult, because of mathematics."
—Steven L. Goldman
And he states that
"The obscurity of the theories of 20th-century physical science from the perspective of the non-scientist public is overwhelmingly a consequence of the forbidding mathematics that has become the language of science. "
—Steven L. Goldman
Science seeks to make sense of the world, but how can it accomplish this goal if the majority of the public cannot make sense of science? Is science, especially physics, now an “esoteric cult” that only a small minority of the population can begin to understand? (One can note here that physicists themselves often do not completely understand all of their theories.) There have been many books written about science for the general public, but these books do not perfectly represent what scientists actually do, or how they think.
Goldman also mentions the uncertainty that the discovery of non-Euclidean geometries caused:
"In the mid-19th century, mathematicians discovered deductively perfect geometries that contradict Euclidean geometry, raising the question of which form of geometry is true of space and severing the uncritical connection between reasoning and reality. (Non-Euclidean geometry: Any logically valid, deductive system of geometry that uses definitions, axioms, or postulates different from the ones used by Euclid.)"
—Steven L. Goldman
How can we be sure that the mathematical theories that we have built up accurately represent reality? If Newton’s clockwork universe, which was the accepted paradigm for many years, did not accurately describe reality, then how can we be sure that our current theories are any better? Of course Newton’s theories of motion are still used to predict the motions of everyday objects. Likewise, quantum theory of the 20th century has been called one of the most productive and useful theories ever. But does this make it true?
"Luitzen Brouwer believed that mathematics is an example of the mind imposing order on experience. There is no necessary connection between mathematics and reality. "
—Steven L. Goldman
Are our mathematical ideas simply beautiful ideas that have no necessary relation to reality? What if reality can be seen from many different angles (to use a mathematical term)? What if the current mathematical perspective is only one of many perspectives? What if it is not the best? If a fish has lived all of its life in water, then how could it imagine any other way of perceiving the world than through the medium of water? Similarly, how could a human being look at the world from a non-scientific, non-mathematical perspective, if he had been saturated all of his life in the current scientific-mathematical worldview?
The final quote I will take from Goldman has to do with symbols:
"Another important development in mathematics in the 19th century was the invention of symbolic logic. From this development, we learned that notation can have a significant impact on our thinking. Simply replacing words with symbols can lead to new insights."
—Steven L. Goldman
In base 10 math, 1+1=2; in base 2 math, 1+1=10; in Roman numerals, I+I=II; in words, one plus one equals two. Of course this is a matter of notation, and changing notations does not change the fact that we are still representing the combination of two things in each case. However, if we use Roman numerals (I, V, X) instead of Arabic numerals (1, 5, 10), mathematics becomes more difficult. Can you add a large number of Roman numerals by hand without first converting them to Arabic numerals? Can you multiply Roman numerals? It is possible, but it is not as easy as calculating with Arabic numerals. Similarly, it is easier for computers to have only two physical states instead of ten, and to deal with base 2 numbers instead of base 10 numbers. Math is heavily dependent on symbols and the meaning of these symbols. Some have compared the learning of math to the learning of a foreign language (although many would argue that this is not a good comparison, and that either math or foreign language is more difficult and that each one uses a different part of the brain). The question is, can one invent or discover a new type of mathematical notation that is more intuitive, easier to understand, and more descriptive of the world?
If only those who understand (the current notations of) higher mathematics can understand the universe, then what about everyone else? Are the mathematically challenged also the reality challenged? Or do current mathematical statements represent reality at all? Are they merely ideas that approximate a certain view of reality? Are we trying to force nature, kicking and screaming, into our mental mathematical molds? Is there another, more universal way of looking at the universe?
Our minds are made out of the same stuff as the rest of the universe, so why should the universe be hard to understand? Are we missing something obvious that will make our understanding of the universe much more intuitive? Are mathematicians barking up the wrong tree? Could a new set of symbols and relationships be what we need to make the universe easier to understand? Or is the universe a foreign place to our minds? Are we strangers in a strange land? I don’t think the universe should seem strange. I think that it is normal for beings in the universe to understand their environments. However, I also think that human beings are currently covered by a veil of ignorance that is present for a particular purpose. Humans will keep banging their heads against the wall (so to speak), attempting to find a “theory of everything,” until this veil is removed.
In the book, My Descent into Death: A Second Chance at Life, Howard Storm writes about the insights that he gained in the spirit realm during his near-death experience (NDE). He was told that
"God wants to give us the power to control matter and energy with our minds, the ability to communicate directly with our thoughts, to travel through time and space by will, to have knowledge by contemplation. The power of these gifts is beyond our wildest imagination, but they will not be ours until we mature spiritually and can use these powers wisely and lovingly."
Can you imagine having knowledge by contemplation? Can you imagine productively exploring the mysteries of the universe by merely thinking about them? Can you give up your paper and pencil, your computer programs? What is the reason that we don’t have these powers, that we still have a veil of ignorance that prevents us from knowing more about how the universe works? I think, to echo what Storm has written, that we don’t have these powers yet because they would be too dangerous in our hands. Humans know how to make all kinds of technological weapons, and much destruction has resulted. Humans have the knowledge and the ability to destroy the earth. Should humans be given the knowledge and the ability to destroy the universe? The key to advancing in the realm of knowledge is to also advance as an entire society in the realm of morality. The key to gaining a greater understanding of the universe is to use wisdom guided by love, and wisdom comes from the loving submission to God’s will.