Philosophy Discussion Forums

Philosophy Explorer wrote:Try this. One lump of clay combined with one lump of clay is just one lump of clay.

PhilX

I know this is an old post, but I cannot but point out the trickery. For one "+" does not mean "combined with" in the context of cardinal numbers. Second, the set whose members are the two individual lumps of clay has a cardinality of 2. Third, a lump of clay and the material it consists in are not the same objects and do not even belong to comparable ontological categories. One is an individual, cohesive, countable object, the other one is stuff that is not countable. "Clay" is a mass term, "lump of clay" is a count term.

There are actually mathematical systems that deal with adding one lump of clay to another lump and getting one lump. I don't have any personal experience with them, but know they exist from some of the math books I have read.

A mathematical truth is only true with respect to its system of axioms, and is not true outside that system.

Fan of Science wrote:There are actually mathematical systems that deal with adding one lump of clay to another lump and getting one lump. I don't have any personal experience with them, but know they exist from some of the math books I have read.

A mathematical truth is only true with respect to its system of axioms, and is not true outside that system.

Yes, but it is not a really interesting observation that different propositions can have different truth values. Sure, "1+1=2" can be interpreted differently in different contexts, but that's semantics. The point is that when you interpret the sentence "1+1=2" in the context of set theory / cardinal numbers, then you are entertaining a specific formal proposition involving specific formal concepts. The question has to do with how many truth values does (or can) that specific proposition have, otherwise we are just playing word games - or at least that is my understanding

It's not word games at all, it's the nature of mathematics. It is impossible to ever refute any mathematical statement regarding "1 + 1 = 2" by referencing empirical evidence. Adding two integers is governed by the axioms of the system. In a certain system "1 + 1 = 2," in a certain system, "1 + 1 = 0," and these systems will not change even if every time someone added one orange to another orange and mysteriously ended up with 5 oranges. This is because the systems of math are based on their axioms and are independent of external evidence. Different mathematical systems, however, can be devised where the "addition" does not involve distinct integers, so that the math can better reflect what is empirically being discussed. However, math will always exist independently of any empirical evidence. This is why Einstein could not refute Euclidean geometry, despite showing spacetime was curved, because Euclidean geometry is based on a specific set of axioms, not on empirical evidence.

Re: Is there a way to refute '1+1 = 2'?

Yes, there is. It is simply

1+1 <> 2.

This whole clay thing: It's not rocket-maths.

One one kilogram lump of clay plus one one kilogram lump of clay equals one two kilogram lump of clay.

read Mathematics ends in contradiction.
gamahucherpress.yellowgum.com/wp-conten ... ssible.pdf
1 number [2] + 1 number[3] = 1 number[5]
ie 1+1=1

Read it last time you posted it. This has all been done before.

gimal wrote:read Mathematics ends in contradiction.
gamahucherpress.yellowgum.com/wp-conten ... ssible.pdf
1 number [2] + 1 number[3] = 1 number[5]
ie 1+1=1

You are comparing apples and oranges.

2 and 3 are different from each other. When you add them together, you get 5, which is different from 2 and different from 3. Yet you claim that they are the same things.

Just because they are numbers.

To you a number is a number is a number.

Well, numbers are different from each other. Different numbers, that is. Ignoring that is ignorant.

Excuse me for not reading all the posts, but please consider the following...

If you can accept the notion that all things [in the Universe] are unique, then how can "2" exist?

Two can exist because it is an abstraction, and we are not using the number to equate to all aspects of a given object. Two dogs means that there are two dogs, not necessarily that the two dogs are exactly the same.

Not to mention that when you get to smaller subatomic particles, they are identical.

FOS, you can not use '2' to prove that 2 exists. Saying that there are two dogs is simply an false idea built in to familiar language. I am suggesting that any object occupying a particular place in space and time is subject to unique set of forces and is therefore, by definition, unique. It might be by "a little bit," but [technically], there is no other object exactly the same. Therefore, 2 can not exist.

It is only through our intellectual laziness and linguistic conventions that we make two possible, no?

That's nonsense. The number 2 is an abstraction, as all mathematical objects are. Being ignorant of the meaning of these mathematical abstractions does not in any way undermine them. It simply means you have not taken the time to learn mathematics before commenting. It's not just math that has this level of abstraction, so do other subjects, like economics. As an example, the amount listed in your bank account is not a reference to any specific, unique, set of dollar bills.

I'll ask again. How is one straight line and another straight line somehow equal to that of a squiggly line?

Spiral Out:

I'll ask again. How is one straight line and another straight line somehow equal to that of a squiggly line?

Have you previously asked that?

The squiqqly line is this symbol '2', yes? And the straight line is this symbol - '1' - when not written by a French person. (They always make them look like '7').

I guess the answer is essentially the same as the answer to this question: Why does a whole load of squiggly lines created by one person on one side of the world cause a whole load of different squiggly lines to be generated by another person on another side of the world?