If you disassemble the engine of your car and put all the parts in any random order in a pile on the ground, it will be obvious that a car engine is much more than just a collection of parts. If this isn’t obvious, do the disassembly experiment and then turn the ignition key and see what happens.

That is not what this post is about, though. Instead I want to point out the difference between sums and sums, or rather the confusions that occur due to the different interpretations that can be applied to the word “sum”.

An arithmetic expression like 1+23+99+42 is typically called a “sum”. But “the sum” of the numbers in that expression is 165 which is a number, not an expression. The expression 1+23+99+42 has distinct parts: these parts are the numbers 1, 23, 42, 99, and the operators that connect those numbers. The sum 165 on the other hand is just one single part all by itself. Or perhaps it consists of 165 parts which are all copies of the number “1”. Or perhaps it consists of the prime factors 3, 5, and 11 multiplied together. The possibilities are endless, and it doesn’t really matter since all these “perhapses” refer to implicit structure, not to anything explicit in the original expression.

The resulting confusion is entirely semantic. When we say “2+2 = 4” we only mean that 2+2 and 4 have values that are identical, not that the expressions themselves are identical. They can’t be identical since the expressions have different structure. Things that are identical can’t have any properties that are non-identical (cf. Leibniz’ law).

The same confusion reappears in many other contexts. One example is “the will of the people”, which is a straight-forward concept when it refers to a collection of varying individual preferences, but it’s something completely different when it refers to a single collectively held preference. The latter is not even well-defined in the general case, due to a mathematical result in social choice theory known as Arrow’s theorem. This has far-reaching consequences for politics and ethics.

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