Ask A Genius 727: Math and Tattoos
Author(s): Scott Douglas Jacobsen and Rick Rosner
Publication (Outlet/Website): Ask A Genius
Publication Date (yyyy/mm/dd): 2022/02/23
[Recording Start]
Scott Douglas Jacobsen: Throughout your life, you’ve been deeply engaged with numbers.
Rick Rosner: Indeed. In fact, I got a tattoo symbolizing my affinity for math, although it’s quite old now. That tattoo dates back to 1988, and as a result, it has become somewhat blurry.
Jacobsen: I’m curious, how would you define a number?
Rosner: To understand numbers, one must begin with the counting numbers, as they are the foundational concept in our understanding of numbers. Over time, we have developed various types of numbers, but any discussion about them inevitably leads back to counting numbers. Discussing numbers entails addressing fundamental existential principles, one of which is non-contradiction. An entity cannot simultaneously be itself and not itself; it must possess internal consistency. Numbers exemplify a high degree of consistency and self-consistency. Basic mathematics allows for extensive exploration without encountering destructive contradictions. However, in more advanced areas of mathematics, such as those involving Gödelian principles, we encounter statements that can never be definitively proven true or false. But these issues lie far beyond the established realms of arithmetic.
Arithmetic has been studied and refined for thousands of years, leading to a general consensus, and possibly even proof, of its self-consistency. In arithmetic, there are no sets of numbers where basic operations yield contradictory results. Regarding counting numbers, the quantity of items in a set is highly subject to the principle of non-contradiction. When dealing with discrete, macroscopic objects, their count yields definite, distinct numbers. These quantities are sharply defined, though we often overlook their precision. When we mention ‘one’ or ‘four’, we refer to an exact quantity—four, not 3.999 or 4.001, but precisely four.
This precision is subject to potential inaccuracies, such as miscounts or anomalous situations, but generally, when counting tangible items like apples, baseballs, or houses, the exactness of their quantities is clearly and accurately defined. For instance, counting three apples or identifying eleven houses on your street demonstrates the precision and non-contradictory nature of simple arithmetic, which underpins its utility and prevalence in existence.
This concept of distinct units becomes less clear-cut at the quantum level. In the quantum realm, the exactness of quantities diminishes. With fewer quantum objects under consideration, and without substantial detection resources, these entities exhibit a degree of fuzziness. For example, a baseball, composed of approximately 10251025 molecules, has a definitive existence with abundant information within those molecules. In contrast, a few subatomic particles in an enclosed space exhibit greater uncertainty. This is exemplified by quantum tunneling, a phenomenon where particles, even within a sealed lead sphere, can unpredictably appear outside of it. This illustrates the inherent indeterminacy and probabilistic nature of quantum particles’ positions in space.
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The likelihood of using a molecule for this is quite slim. A molecule is extremely unlikely to escape from a lead ball through tunneling, due to the minuscule probability and uncertainty in its position, even over the entire lifespan of the universe. However, a photon or an electron, depending on the thickness of the lead sphere, could have a different outcome. You could, theoretically, construct a lead sphere thin enough that an electron inside it might have a one in a billion chance of tunneling out within a year. You can make the sphere as thin as desired.
Moving on, a vast number of objects exhibit fuzziness; they lack definitive existence. Yet, we can create a defined world by assembling enough indefinite particles that define each other. This is the world we inhabit, a highly defined world composed of approximately 10851085 particles that mutually define each other. The primary massive, discrete objects in the universe are stars. With about 10111011 galaxies, each containing roughly 10111011 stars, we have a total of 10221022 stars, each comprising about 10601060 particles.
Considering planets, the Earth has significantly less mass than the Sun. While I initially thought the Earth might have 1/100th the mass of the Sun, that seems inaccurate. The Earth’s diameter is about 1/100th that of the Sun, which, assuming equal density, would imply the Earth has 1/1,000,000th the mass of the Sun. However, Earth is approximately five times as dense as the Sun, leading to a calculation of about 1/200,000th the mass of the Sun. Let’s approximate it to 1/1,000,000th, meaning a planet still has more than 10501050 atoms or protons, an immense number supporting a high degree of definiteness.
The universe is composed of a vast array of highly definite, existent objects, ranging from a baseball with 10251025 particles to a planet with 10551055 particles, and up to a star with 10621062 particles. To achieve existence, a large number of particles must come together over time. This leads to another principle I believe to be true, though without much proof beyond our universe’s existence: there’s no upper limit to the number of things, short of infinity. Existence arises through finite processes, so infinity is unattainable, but anything short of infinity is conceivable.
Therefore, however large our universe is, there could be universes with the number of particles in our universe squared. It’s conceivable to have a universe with 1017010170 particles or even 1025510255 particles. In the realm of all possible universes, I suspect there’s no limit to that exponent on the number of particles.
[Recording End]
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