In the middle of studying for my last exams, I could not help overhearing some students in the library complaining about their mathematics course. Their complaints were comprehensive; the professor was difficult, the notes were lackluster, and the homework was too tedious. After the course-specific discussion ended, one of the individuals in the group made an interesting claim, namely that “math is useless because it is made up. What makes 0, 0 and 1, 1? I will just change the rules – 1+1 is now equal to 3.”
While this individual’s question was specific to mathematics, it could be generalized into the following form: “How did we come to accept common explanations for the way the world functions?” This piece is my answer to that question. The ideas shared here are derived from David Deutsch’s The Beginning of Infinity and I have focused on thoughts from the book that may be of practical use to those of you reading.
The first component in constructing a response to the question “How did we come to accept common explanations for the way the world functions?” is to identify the difference between good and bad explanations. An explanation is more than a correct prediction that a particular outcome will occur. Rather, good explanations illuminate the underlying structure of how an outcome occurs, and consequently, one cannot simply alter aspects of a good explanation because it would no longer be valid. For instance, consider Deutsch’s explanation for the existence of seasons (spring, summer, fall, and winter):
“It is that the Earth’s axis of rotation is tilted relative to the plane of its orbit around the sun. Hence for half of each year, the northern hemisphere is tilted towards the sun while the southern hemisphere is tilted away, and for the other half, it is the other way around. Whenever the sun’s rays are falling vertically in one hemisphere, they are falling obliquely in another.”
Each part of this explanation has utility and altering any piece of the explanation would cause it to “break.” It explains why seasons change, why seasons occur in different months across the hemispheres and explains the role the sun’s rays play in heating the surface of the Earth. For instance, if I change the explanation above and state that, the Earth’s axis of rotation is not tilted, but vertical, then this would imply that seasons would not occur in different months across hemispheres (which is clearly false). In contrast, consider the Nordic explanation for the existence of seasons:
“In Nordic mythology, seasons are caused by the changing fortunes of Freyr, the god of spring, in his eternal war with the forces of cold and darkness. Whenever Freyr is winning, the Earth is warm; when he is losing, it is cold.”
This is a poor explanation for seasonality as one can change its components without a loss in its explanatory power. If we assert that when Freyr is losing the Earth is warm, and when he is winning it is cold, it does not alter the existence of the seasons – they still exist, but the narrative surrounding the existence of these seasons has shifted. Additionally, this explanation would fail in a testable sense because it implies that seasons are constant across the Earth at any point in time.
Good explanations are essential to the advancement of civilization because they facilitate the creation of knowledge. I have taken a liking to Deutch’s definition of knowledge which he defines as information that, once physically embodied in a suitable environment, tends to cause itself to remain so. Knowledge that still exists today remains so because it had utility for individuals that used it before us. The creation of tallies enabled our ancestors to keep track of quantities in their possession. Farmers could tally the number of crops they harvested – I, II, III, IIII, and so on – as well as the amount of livestock the possessed. By taking a step further, tallies could be generalized into numbers by abstracting tallies (one, another one, another one) into counts (1, 2, 3, 4).
To illustrate the utility of numbers, observe that the number 4 encodes information for farmers that is embodied in a suitable environment. Specifically, it represents the knowledge of the number of crops or livestock a farmer has in his or her possession. The creation of numbers facilitated the development of arithmetic, whereby individuals could add and subtract items in their possession, providing them with a more efficient accounting method than tallying. Consequently, addition provided farmers with a magnificent tool.
Consider a farmer who has an enormous amount of livestock in his possession and who recently obtained new offspring. Rather than being forced to conduct the tallying process over again (one, another one, another one), he can simply take his existing count and add 1 to it, whereby this new count is an accurate representation of the amount livestock in his possession. 1+1 is not 3 because it provides its users with incorrect knowledge. If a farmer had one cow and received another, he now has one and another one, for a tally of II and a count of 2. This arithmetic system has been passed on over time due to its usefulness, justifying the definition of knowledge used above – it tends to remain in a suitable environment once it is physically embodied in one.
In fact, our ability to create knowledge via creative thought has enabled humans to escape the constraints of biological knowledge creation. Natural selection is useful in creating knowledge – genes encode information that is useful in perpetuating the existence of a species. However, because humans can create useful explanations for phenomena, knowledge is created as fast as we can subsidize it. Once knowledge is manifested in a suitable environment, it is passed down to the next generation, whereby their own creative thought can be used to enhance the knowledge that existed before them.
This process of creating knowledge is boundless if we assume that many of our explanations can be improved upon. If we are consistently questioning the validity of explanations that currently exist, humans should be motivated to continuously improve upon our models of how the universe works. Additionally, the belief that we can always enhance our explanations is an important one, because that optimism is essential to progress. Since we cannot be certain as to what we will discover in the future, an optimist believes consistent progress can be made.
For those that question whether humans can develop certain technologies, consider the following: Good explanations exist because the world around us is sensible; in essence, it is possible to explain. Since nature has structure, then we can create explanations that define how nature operates, conditional upon developing the right knowledge. Once knowledge is created, technology is subsidized with this knowledge. However, if we are incapable of creating or applying certain technology, then there must be an explanation for that since nature has structure. Thus, “everything that is not forbidden by the laws of nature is achievable, given the right knowledge.”
After considering the logic outlined above, it is difficult to not feel optimistic about what humans can accomplish. The mathematics, technology, and theories that we have developed exist because they have proved useful to us. If we continue to foster creative thinking and leverage the knowledge that was created before us, we will continue to enhance our understanding of the world – one good explanation at a time.