Herb Silverman is the Founder of the Secular Coalition of America, the Founder of the Secular Humanists of the Lowcountry, and the Founder of the Atheist/Humanist Alliance student group at the College of Charleston. Here we talk about the meaning of numbers and delve somewhat into the notions, or the formal mathematical concepts, of mathematical objects, and more.
Scott Douglas Jacobsen: What makes a number, a number? How does this relate to the discrete or continuous nature of the world?
Professor Herb Silverman: A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, …., which were extended to take in 0 and the negative integers. This later included rational numbers (fractions), irrational numbers (real numbers that are not rational) like pi and the square root of 2, and complex numbers like the square root of -1.
Besides their practical uses, numbers have cultural significance throughout the world. For example, in Western society, the number 13 is regarded as unlucky. Some people also believe in numerology, which attributes a divine or mystical significance to numbers. One such example, espoused by many Christian fundamentalists, is fear of the number 666, which they refer to as the Mark of the Beast. Numerology is also associated with the paranormal and astrology. Of course, numerology is a pseudoscience, a superstition that uses numbers to give their subject a veneer of scientific authority.
The question about the discrete or continuous nature of the world is an important one, with many implications. By discrete, we mean something we can count and that can’t be further divided. An example would be the number of students in a class. Continuous is the opposite of discrete. It can always be divided into finer levels.
How old are you? I could say that I’m 77 years old, but it would be a lie. Though I recently celebrated the seventy seventh anniversary of my birth, it would be impossible to say how old I am. The same with you. As of this writing, I’ve been alive approximately 77 years, 20 days, 6 hours, 7 seconds, 5 milliseconds, 3 nanoseconds, 1 picosecond (a trillionth of a second)…and so on. I would be simply 77 if time were measured only in years, which it is not.
We treat time as if it is continuous, not discrete. The same with height and weight. It is impossible to know exactly how tall you are or how much you weigh. Similarly, when we see a movie the time it portrays looks continuous, though we know it was made with discrete scenes. In classical and quantum mechanics, time is treated as continuous. Otherwise, the physics would not be applicable. Nevertheless, we don’t really know if time (or space) is discrete or continuous.
Max Planck was a theoretical physicist who did revolutionary work in quantum theory. Planck time (approximately 10 to the — 43 seconds) is the shortest possible time interval that can be measured. With its associated Planck length (approximately 10 to the -35 meters), the Planck time defines the scale at which current physical theories fail. On this scale, the entire geometry of spacetime as predicted by general relativity breaks down. It is possible that time might be advancing forward in tiny but discrete time intervals, or time might be continuous. We just don’t know. The same is true about space.
All scientific experiments and human experiences occur over time scales that are many orders of magnitude longer than the Planck time, making any events happening at the Planck scale undetectable with current scientific knowledge. The smallest time measurement has been approximately 10 to the -21 seconds. Before Planck time all matter, energy, space, and time are presumed to have exploded outward from the original singularity (a point or region in spacetime in which gravitational forces cause matter to have an infinite density, associated with black holes). Nothing is known of this period. Looking backward, the idea is that back beyond a Planck time we can make no meaningful observations within the framework of classical gravitation.
Rather than thinking of discrete intervals, we should recognize that it is not possible for us to make a measurement of length or time smaller than Planck values for length and time. Physics can say nothing about shorter intervals, which is why we can’t go back to time zero of the Big Bang. I think that someday more will be discovered, but not in my lifetime and maybe not in yours. It will take a serious amount of time for humans to understand time.
Jacobsen: Thank you for the opportunity and your time, Professor Silverman.