The Abstract World
Mathematics and logic require abstracting away real-world details to create a perfectly logical system, but one must be cautious when applying those results back to reality.
THE IDEA
THE ABSTRACT WORLD
Most real objects do not behave according to logic. I don’t. You don’t. My computer certainly doesn’t. If you give a child a cookie and another cookie, how many cookies will they have? Possibly none, as they will have eaten them.
This is why in mathematics we forget some details about the situation in order to get into a place where logic does work perfectly. So instead of thinking about one cookie and another cookie, we think about one plus one, forgetting the “cookie” aspect. The result of one plus one is then applicable to cookies, as long as we are careful about the ways in which cookies do and don’t behave according to logic.
Logic is a process of constructing arguments by careful deduction. We can try to do this in normal life with varying results, because things in normal life are logical to different extents. I would argue that nothing in normal life is truly entirely logical. Emotions, missing information, limited processing capacity, and randomness can complicate the attempt.
So in order to study anything logically we have to forget the pesky details that prevent things from behaving logically. In the case of the child and the cookies, if they are allowed to eat the cookies, then the situation will not behave entirely logically. So we impose the condition that they are not allowed to eat the cookies, in which case those objects might as well not be cookies, but anything inedible as long as it is separated into discrete chunks. These are just “things”, with no distinguishable characteristics. This is what the number 1 is: it is the idea of a clearly distinguishable “thing”.
This move takes us from physical objects to an abstract model whose assumptions we can state and examine.
When to use it
Solving a math word problem by stripping away irrelevant context to focus on the numbers.
Creating a scientific model of a physical system, simplifying it to manageable equations.
Making a decision using a logical framework while acknowledging emotional factors are omitted.
Designing a computer algorithm that assumes perfect inputs, then handling real-world edge cases.
FROM THEORY TO EVERYDAY LIFE
In practice
Cookies and Counting
If you give a child a cookie and another cookie, the child might eat them, so the number of cookies changes. Mathematically, we abstract away the child's eating and treat the cookies as unchanging units, so 1 + 1 = 2. This works only if we assume the cookies are not eaten.
Geometry of Land Surveying
A surveyor may use a flat polygon to estimate the area of a plot with irregular terrain and boundaries. Whether that model is adequate depends on the area being measured and the precision required.
A MOMENT TO REFLECT
Put it to work.
What real-world details do you tend to ignore when applying a mathematical or logical model?
Can you think of a situation where applying an abstract model led to a mistake because real-world behavior broke the assumptions?
How does abstraction help you think more clearly, and what are its limitations?
CONTINUE THE CONVERSATION
Source texts
- The Art of Logic How to Make Sense in a World that Doesn’t (Eugenia Cheng)