MathSci Problems
The Limits of Definitions
Definitions are important. In science, units like the kilogram let engineers work with precision. In law, a defined “contract” makes it possible to decide what counts as binding. Even day to day, words like “friendship” or “success” influence how we see ourselves and others. But most things in life don’t come with sharp edges. Definitions give us structure, but not everything fits inside the lines.
Colour’s a good example. Where exactly does green become blue? We can invent words like “turquoise,” or use HEX codes to be more precise, but we’re still drawing lines through a continuous spectrum. The same thing happens when we try to define emotions, identities, or values. The cleaner the label, the more of the reality it leaves out.
Tax brackets show this in another way. In Canada, the federal tax rate jumps from 20.5% to 26% once your income passes $114,750 (as of the time I wrote this). So someone earning $114,000 and someone earning $115,000 fall into different categories, even though their financial situations are basically the same. The system works for bookkeeping, but it doesn’t reflect how people actually experience money.
In biology, species don’t always split cleanly. Ensatina salamanders in California form a ring: neighbouring groups can interbreed, but the ends of the ring can’t. Are they one species, or several? Dogs and wolves can produce fertile offspring and share almost all their genes, but we label them differently.
Fuzzy logic is a potential solution to the limitations of strict definitions. Instead of assigning things a “yes” or “no,” it gives them a percentage. A room might be 70% warm and 30% cool. That kind of thinking shows up in smart thermostats, which adjust gradually depending on how close the temperature is to the goal. In medicine, it helps describe symptoms that suggest a condition without fully matching the definition.
There’s also a philosophical way to rethink definitions. It doesn’t just ask how we define, but where meaning comes from in the first place. Linguist Ferdinand de Saussure argued that words mean what they do by contrast. We understand “night” because it’s not “day.” Meaning isn’t in the word itself, but in its relation to other words.
Post-structuralist thinkers like Derrida pushed this further. They argued that meanings depend on the situation. “Justice” in a courtroom might mean due process. In a protest, it can become a call for equity or resistance. In a relationship, maybe it’s fairness or closure.
Strangely enough, this idea isn’t limited to modern theory. A similar idea appears in Madhyamaka Buddhism. It teaches that things don’t have fixed essences. A table is only a table because of how we use it and what we agree to call it. In another setting, the same object might be a bench, a piece of firewood, or a shelf. When we cling to definitions too tightly, we start mistaking the label for the thing itself. The point is to use them when they help and let them go when they don’t.
Definitions help us move through a complex world, but they always leave something out. Maybe the real challenge is knowing when to trust the map, and when to look up and see where we actually are.
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