Editing Ontology:Q2681 (section)

== Usage notes ==

It's been popular in some strains of traditional philosophy to try to analyze mathematics through the notion of "truth" and "kinds of truth". Philosophers will try to list out every kind of statement they would consider to be answerable with the boolean value of "true", and then separate out "empirical" truths from "logical" truths to try to assign some specific kind of truth to mathematics.

It is arguable this approach to understanding the relationship between mathematics and what statements are true is simply incorrect. To some small extent, all true statements are social constructs. Even when we remove statements from language <i>per se</i>, every statement is still composed of signifiers in the [[redlink - Q?? ontological model|ontological-model]] sense — every object or process in a statement must receive some kind of mathematical or logical definition which is not always perfectly objective or accurate, and this will always shape our process of parsing what statements about it are logically true. For one example, scientists can model fundamental particles using wave functions, and all calculations which produce facts about how wave functions interact will be based on that possibly-flawed definition. We can never actually say with absolute certainty that a statement we logically extrapolate from mathematical data structures or definitions is accurate to material reality, and if this is our standard for what is most clearly "true", then it is not possible to say that "materially true" and "logically true" are even the same thing. One might as well assign them different words: one is [[redlink - facticity|factical]], the other factual, one <em>is</em>, the other <em>proceeds</em> or <em>follows</em>. One <em>is</em>, the other <em>would be</em>, but never <em>is</em> or <em>was</em>. This distinction is basically the same as the distinction between the material world and the hypothetical. A good hypothesis about reality predicts the world, but there is a seemingly infinite range of possible hypothetical statements which may have nothing to do with reality, and which at certain points may even have less and less to do with themselves.

Say we decide to test arithmetic using real objects. We take two apples and put them next to two apples. There will be four apples. The hypothetical model of integer arithmetic describes the apples, as does the hypothetical model of set theory. Set theory is not actually the same model as integer arithmetic; they are two different internally-consistent models which happen to connect with each other and derive from each other. Now, say we take two ice cubes and two lit matches and put them next to each other. Likely, both the ice cubes melt after a while, but the flames will still be there unless they were held close enough to the ice cubes to fizzle out. Already there are two possible interactions: one flame takes out one ice cube while the ice cube reciprocally puts it out, or one flame takes out one ice cube while continuing to burn. There is also a third possibility: we put a match on the ice cube and the flame goes out but the ice cube stays intact. Does two flames plus two ice cubes equal four ice cubes? Does it equal four flames? It probably doesn't even equal four intact objects, but there is not just one possible outcome in terms of non-intact objects. If the ice cubes melt, we could consider the flames to be negative numbers. In one case two flames plus two ice cubes is similar to the equation 2 + (-2) = 0. In the case the ice cube wins, we could say the situation is something more like 100 + (-2) = 98. But the case where the flames stay burning and the ice cubes melt after a long time is more complicated. Understanding this case would require something much closer to an actual Newtonian physics equation. How many joules of energy are released by the match? How many are transferred through the air? How many joules of heat can water absorb? How much heat is removed from an ice cube to turn it to ice compared with the temperature it was at the moment it went into the freezer, or the temperature it would be at after it came out and melted on its own? Water and ice do not seem complicated in daily life, and yet modeling the physical structure of removing heat to turn water molecules into a crystalline lattice is significantly more complicated than describing putting two apples next to two apples. (This is thanks to Avogadro's number, the basic order-of-magnitude increase in number of objects between an atom and a gram of carbon, water, or apple.) Thus, anyone who interacts in any way with chemistry or the arrangement of atoms will be forced to admit that mathematics, particularly arithmetic, is a simplified model of reality which is [[redlink - deterministic brushstrokes|arbitrarily applied by human beings]] to the set of situations it is shown to accurately model. Human beings never fully have the ability to make something be true just because it is defined to be true. If that thing has any interaction with material reality which shows it to be a mere simplified model, then material reality will decide which things are true.

With that said, there are both positive and negative philosophical ways to interpret this observation. One could take it to mean that there must be a level of [[redlink - objectivity in meta-Marxism|objective truth]] that is more important than the human opinions or experiences we embody in mathematics, but another interpretation is that when we realize that mathematics is arbitrary, we see that there are multiple possibilities and we can open up our creativity to search for new forms of mathematics or new ways to apply mathematics to model the world. Mathematics being partly arbitrary could be seen as the statement that fields of mathematics can be actively invented and improved in order to make them more useful or accurate.


[[Category:Meta-ontology ontology]]