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

Use this forum to discuss the philosophy of science. Philosophy of science deals with the assumptions, foundations, and implications of science.
Hereandnow
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### Re: Is there a way to refute '1+1 = 2'?

Just a quick reminder that if you think that you can refute mathematics you're (nothing personal) basically out of your gourd. For, as has been famously made clear by many, when you bring the rules of math and logic under review, you are reviewing with those very rules you are trying to critique, and this instantly invalidates whatever you say. In other words, you, the cognizing agent, can never "get behind" the logical tools you employ to observe those very logical tools themselves. '1=1+2' cannot be "refuted" any more than modus ponens, the principle of identity or Demorgans theorem, for to do so you would have to step out of logic itself, and that would require another system of "cognition", or whatever it would be called, which can't even be imagined.

What you can do is speak nonsense in the Wittgenstinian therapeutic manner: No mathematical truth is absolutely sustainable, for no absolutes can be reasonably stated, since meaning can only occur within a body of contingently organized, truth conferring propositions. Therefore, '1=1+2' is demoted from its high horse of apodicticity (though this cannot be actually demonstrated) to, as Rorty and others would have it, a mere social contigency. (How social? Tough question. As I see it, after reading a paper by Herbert Meade and others, the mind is an evolved entity, and its structural features, like cognition, are essentially social, given that language is a social function.)

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

Hereandnow wrote:'1=1+2' cannot be "refuted"
(I'm assuming you meant "1+1=2")

So then why is a straight line and another straight line the same as a squiggly line?
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Logic_ill
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### Re: Is there a way to refute '1+1 = 2'?

Sure, by using other symbols that probably mean the same thing anyway. But another being may not view a unit as one, but as composed of various parts, due to its sensory perception. So, one rock, for example, may be perceived as various pieces, and the concept would not be grasped.

Even the line, which represents one to humans, may not seem like a line to other "intelligent" beings.

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

Spiral Out: So then why is a straight line and another straight line the same as a squiggly line?
When is a straight line not a straight line and a squiggly line not a squiggly line? When these have symbolic designations.The signifier, the lines, are arbitrary. The concept, or the logical "sense" is what is in play.

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

The concept behind mathematics (1+1=2) is an entirely arbitrary construct which was created and (semi)voluntarily agreed upon by beings with a fundamentally limited capacity for understanding the nature of an objective reality.

It is quite easy to refute the idea that 1+1=2 simply by disagreeing with the provisional definitions of the terms.

The only thing that is irrefutable is the objective truth that an object next to another object is as it appears, which doesn't necessarily correspond to the concept of 1+1=2.
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Hereandnow
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### Re: Is there a way to refute '1+1 = 2'?

Spiral out: The concept behind mathematics (1+1=2) is an entirely arbitrary construct which was created and (semi)voluntarily agreed upon by beings with a fundamentally limited capacity for understanding the nature of an objective reality.

It is quite easy to refute the idea that 1+1=2 simply by disagreeing with the provisional definitions of the terms.

The only thing that is irrefutable is the objective truth that an object next to another object is as it appears, which doesn't necessarily correspond to the concept of 1+1=2.
Objective reality? What's that? Logicality is not arbitrary; but then, if your claim is that 1+1=2 can be valid only if compared to Objective Reality (the state of affairs that that is REAL, non-contingently the case) then your refutation must be cast in terms outside of contingency in order to speak reasonably of such a thing. Note that the utterance "The concept behind mathematics (1+1=2) is an entirely arbitrary construct" is a meta-logical one. Where is basis for this?

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

It is a question of which tools are appropriate. Any logical argument is built on axioms and the axioms appropriate in one area are not appropriate in another.

For example, what is true in Euclidean geometry is not true in three dimensions, so since no object exists in a Euclideian universe (i.e in only two dimensions) in what sense is a Euclidean proof 'true'?

Nor do the rules of geometry apply to maths; a fact that troubled philosophers from ancient times. If we claimed both maths and geometry deliver something called 'the truth', then surely it ought to be the same truth?

Similarly, add one pile of sand to another pile of sand and you will not make two piles of sand. If you add John to Jane you do not get 'two John' or 'two Jane'. Maths deals only with abstractions, not specific objects. So again, since the world contains only specific objects rather than abstractions, in what sense is maths true?

On its own, unapplied logic or maths cannot be 'refuted' but that is only because it does not assert anything. But if we attempt to use logic then we must insert propositions; these propositions may be true or false and that will be what determines whether the conclusion is going to be true.
"The concept behind mathematics (1+1=2) is an entirely arbitrary construct" is a meta-logical one. Where is basis for this?
I'd argue it should be the other way round. Our lives consist of experiences, we may reflect on those experiences, then we may reflect on abstractions of those experiences. That is how we arrive at maths, so if anything was 'meta' it would surely be the maths!

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

Londoner ;

I'd argue it should be the other way round. Our lives consist of experiences, we may reflect on those experiences, then we may reflect on abstractions of those experiences. That is how we arrive at maths, so if anything was 'meta' it would surely be the maths!
Unless that particular "meta" you have in mind is "about" the very rules you are using in your critique.

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

Unless that particular "meta" you have in mind is "about" the very rules you are using in your critique.
You miss my point that there is no one set of 'rules' that we use.

If we could solve problems by maths or logic - then there would be no problems! We could settle every question using a pocket calculator.

But in practice we often don't agree either what is true, what truths are relevant, or which 'rules' we should apply to them.

The only thing numbers concern themselves with are quantity. That is only one of many categories of understanding. I will illustrate:

Imagine a chair. Now imagine another chair - this chair looks just like the first one. Using 'number' we can say 'there are two chairs'. But we can equally say 'both chairs are the same'. Both descriptions are true.

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

Londoner:

You miss my point that there is no one set of 'rules' that we use.

If we could solve problems by maths or logic - then there would be no problems! We could settle every question using a pocket calculator.

But in practice we often don't agree either what is true, what truths are relevant, or which 'rules' we should apply to them.

The only thing numbers concern themselves with are quantity. That is only one of many categories of understanding. I will illustrate:

Imagine a chair. Now imagine another chair - this chair looks just like the first one. Using 'number' we can say 'there are two chairs'. But we can equally say 'both chairs are the same'. Both descriptions are true.
How do you "know" both descriptions are true? You see, even at your first utterance, you invoke the logic embedded in your language. These are not empirical rules that you can take or leave; they are apriori, and are forced to use them. Take a simple assertion: the cat is on the mat. Certainly, there are features of this that are matters of convention. That 'cat' is the noise made to designate those fluffy four-legged creatures, or that the verb follows the subject (unlike the Korean counterpart), or the prepositional phrases are constructed in such a manner, etc. But underlying these conventions there is a foundation of logic, as is found in symbolic logic: modus ponens, disjunctions, hypotheticals and the rest. Unless you can use these, you simply cannot speak. You cannot, as has been put many times, ever "get behind" these; that is, assume a critical perspective that does not employ them. They are what thought is. Since you can't do this, then talk about its possibility is nonsense.

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

All of these rules you're referring to are the rules Humans created in order to effectively function within our environment and with each other, they aren't the rules of the environment itself. Mathematics is not a fundamental component of the environment, therefore the rules of mathematics (as arbitrary) can be refuted outside of that specific mathematical construct, especially since there is far more outside of the mathematical construct than there is within it.
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Hereandnow
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### Re: Is there a way to refute '1+1 = 2'?

Spiral Out: All of these rules you're referring to are the rules Humans created in order to effectively function within our environment and with each other, they aren't the rules of the environment itself. Mathematics is not a fundamental component of the environment, therefore the rules of mathematics (as arbitrary) can be refuted outside of that specific mathematical construct, especially since there is far more outside of the mathematical construct than there is within it.
Errrr...and what particular mathematical construct would that be? You can say that we invent the manner in which rules take shape and are applied in the world, as we do when we conceive of a banking system or design an automobile. But you CAN"T say we invented the very structure of logic itself.

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

Hereandnow wrote:You can say that we invent the manner in which rules take shape and are applied in the world, as we do when we conceive of a banking system or design an automobile. But you CAN"T say we invented the very structure of logic itself.
Yes, I most certainly can, especially if that very structure of logic is 1+1=2. If the very logic itself is what is merely described by 1+1=2 then that very structure lies within the realm of the objective reality that you have previously admitted as not being within the realm of mathematical logic.

There still exists a valid refutation of mathematical logic due to the fact that the two realms cannot be reconciled, much like GR and QM.
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Hereandnow
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### Re: Is there a way to refute '1+1 = 2'?

Spiral Out: Yes, I most certainly can, especially if that very structure of logic is 1+1=2. If the very logic itself is what is merely described by 1+1=2 then that very structure lies within the realm of the objective reality that you have previously admitted as not being within the realm of mathematical logic. There still exists a valid refutation of mathematical logic due to the fact that the two realms cannot be unified, much like GR and QM.
Very good. But 1=1+2 does not describe its own logic. It shows it, or expresses it. But to describe logic requires a vantage point that is not logical. it would be like describing a ride on a Ferris Wheel using more Ferris Wheel rides. Description needs another medium of symbolic possibility that is alogical (which is impossible to conceive). But to ride a Ferris Wheel--this is quite different: The ride, if you will, speaks for itself; it is pure engagement. This is what the logical intuition of 1=1+2 is.

And this is why true refutation is impossible.

Regarding the valid refutation of mathematical logic: The two realms are mathematics and logic? And I presume you have in mind some sort of reductio the two generate in some theorem. Just note that as this reductio is observed, the observer, you, are "engaged" in the logic (which is not what you have, but what you are) that generates the intuited contradiction. The best you could say is that thought and its logic is puzzlingly contradictory at times. If you think this shows that logic is just a house of cards that collapses 9as it does with such performative contradictions as "This sentence is false"), and takes 1=1+2 with it, then you would have to admit that the very logic you are employing to make this claim is duly refuted as well and your refutation is equally indictable.

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

Me: Imagine a chair. Now imagine another chair - this chair looks just like the first one. Using 'number' we can say 'there are two chairs'. But we can equally say 'both chairs are the same'. Both descriptions are true.

How do you "know" both descriptions are true? You see, even at your first utterance, you invoke the logic embedded in your language.
No, I will know they are true not through logic but because I look at the chairs and see that there really are two of them and also that they are alike.

If I was suffering from double vision, i.e. there was really only one chair, then my description would be wrong. Logic would not come into it.
These are not empirical rules that you can take or leave; they are apriori, and are forced to use them. Take a simple assertion: the cat is on the mat. Certainly, there are features of this that are matters of convention. That 'cat' is the noise made to designate those fluffy four-legged creatures, or that the verb follows the subject (unlike the Korean counterpart), or the prepositional phrases are constructed in such a manner, etc. But underlying these conventions there is a foundation of logic, as is found in symbolic logic: modus ponens, disjunctions, hypotheticals and the rest. Unless you can use these, you simply cannot speak.
'The cat is on the mat' can be reduced to my pointing at the two objects.

But logic etc. is entirely unconnected to objects like cats and mats.

In symbolic logic we use symbols rather than words precisely because it is unspecific; any proposition would do, whether it is really true or false doesn't matter. In logic we are only interested in the connections, the symbols are simply place markers for whatever truth values we decide to give them.

It doesn't assert any fact about the world; '1 + 1 = 2' does not assert 'there are two apples'. If it did then it would be wrong! There would still be this green apple over here, and that red apple over there in the identical state to before they got 'added'. Maths only works if we move away from actual objects and deal with abstractions. 'John plus Jane' does not make 'two'. We first have to abstract them into 'people', i.e. we must abandon any interest in the truth of any fact about John or Jane, i.e. whether these people exist etc..

The question we are discussing is 'Is there a way to refute '1+1=2'?'My answer is that there is nothing to refute, because it doesn't refer to anything. We might as well be asking; 'Is there any way to refute concept'? or 'Does it exist?'