People tend to assume that mathematics is point where we can all agree. That is to say, we can all agree that 1+1=2, regardless of our philosophical standing. Again, that is to say that mathematics is a philosophically neutral ground. I would like to examine this. (The main focus on my thread will begin with the epistemological and metaphysical claims. I’ll try to give a brief introduction, I don’t know how much it’ll help, though…)
This may seem absurd. It may seem obvious that everyone believes that 1+1=2. However, it might surprise you to learn that not everyone does. Radical monists would state that everything is one. Some versions of Hindu think that plurality is illusion, so on the ultimate level of being, everything is again one. This means 1+1=1.
This shows us that even the simplest arithmetical truths can be sustained in a world-view which acknowledges an ultimate metaphysical plurality in the world. It also must be that there is an ultimate metaphysical unity, since we have “sames.” Two sticks remain sticks even though there are two. In different equations, the number two is the same number, but at different times.
So beginning this thread, we have the problem of the one and the many. We will not talk about that here, though. But to be sure, it is an important problem, and it’ll be raised later.
Some mathematicians do not agree on the validity of certain proofs. For example, intuitionists do not accept the law of excluded middle or the proof by reduction ad absurdum. The human mind is where truth rests. Mathematics is concerned with mental constructions. Therefore proofs that we cannot grasp do not exist. If you aren’t aware of this, and would want an example, ask. Anyhow, this poses a problem.
There are disputes in geometric truth, truths of analysis, and even mathematical existence. I won’t go into this for the sake of a very long thread, but if you want, I can go into it as much as I know.
Anyway, mathematics seems to have at least two problem areas. They are epistemology and metaphysics.
Epistemology
Here we discuss how we come to the knowledge that 1+1=2. Do we know it a priori, or internally, without experience, or do we know it a posteriori, or externally, derived from experience. Plato claims that we gained it by reminiscence or introspection. Bertrand Russell says logical arguments. John Stuart Mill would say from repeated experience. A. J. Ayer says it’s a linguistic convention and not real “knowledge.”
A priori. 1+1=2 is a universal truth. Why should 1 stick plus 1 stick make 2 sticks, though? It might seem ludicrous, but this is a real problem. If we know it, internally, and without experience, why should the external world be contingent and offer us repeated examples of this. If the external world is made up of completely chance matter, why is there a connection? Why don’t sticks disappear and appear randomly while we are counting them? If the outside world has a degree of regularity mixed in with chance elements, why should that regularity coincide with the internal mathematical expectations of the human mind? When you make the Cartesian separation of a priori[/i[ and a posteriori you can’t get them back.
Further, if mathematics is a priori, why do paradoxes arise.. . especially in the form if contradictions… (See Burali-forti’s paradox, or Bertrand Russell’s paradox) or in the form of various counter-intuitive results (Peano’s space-filling curves, Lowenhein-Skolem theorem)… These paradoxes seem to show that saying that mathematics is a priori is not really reliable…
A posteriori. Because of the difficulties of the a priori explanation of mathematics, we have the a posteriori explanation. Mathematical knowledge is inductive. See my induction thread also for better knowledge of induction.. We know that 1+1=2 from repeated examples of one object plus another one making two objects. However, we haven’t seen 234343 objects plus 2343 objects making 236686 objects (I hope my math is correct lol). But we have generalized. But to say this, hints at the a priori answer. Further, why don’t we generalize that 23+1=24 instead of 23+1=25? You can really only say because you have generalized from other generalizations. You do this because you have generalized from previous experience of generalizing, etc…. See my thread on induction. Further, at some point, you might say that this is the way the human mind operates. We then fall into the a priori explanation and it’s problems.
Conventional answer. Mathematicians seem to accept this answer. Mathematics is in some sense a convention of the human language. It isn’t knowledge at all. 1+1=2 because we have agreed to use the words one and two in that way (see Wittgenstein). To say 1+1=2, we are saying A=A in a roundabout way (A.J. Ayer).
These seem like the a priori answer. Why should mathematics prove useful in dealing with the outside world? If it is pure convention… If the conventions are chose because they are useful, we have the a posteriori answer. The conventional answer isn’t really an answer. It’s only a shifting of the question.
Reading Godel’s proof would help you understand the difficulties of all of these answers. I won’t go into it here, because of the massiveness of this thread already.
Metaphysical
The one and the many. 1+1=2 means something. If someone told me that 1+1=2, say a baby for example muttering his first words, I wouldn’t believe that he understood that until he could demonstrate it (do word problems, show the symbols on a chalkboard, etc). So it would be safe to say that we can’t know 1+1=2 until we know many things in relation to that truth. So we are concerned with many experiences and truths in relation to that one.
Modern linguistic theory has pointed out that we can’t understand these symbols without some specification of it’s contrast. 1+1=2 only has meaning as part of a greater system (a plurality). 1+1=2, 1+2=3, and that 1+1=3 is false, etc. We should also understand it on a variational range. 1+1=2, one plus one equals two, I + I = II. We should also understand it distributed in larger units of linguistic behavior and human behavior (proofs using this, word problems, used in calculating your bills, figuring out prices, etc…)
We have to show the unity and stability in this vast amount of truths and experiences though. Some people ask, how can we know anything without knowing everything? This is a valid question. As we have seen, we must know the larger context in which the proof is embedded until we can understand the proof. But we don’t know all of the larger context. How can we? Therefore the next question is how do we not know that the next thing we discover will upset our knowledge.
Physics has changed during the Newtonian revolution, the Einsteinian revolution, and the quantum revolution. Mathematics has revised itself, too, at times (discovering irrationals by the Pythagoreans, discovering contradictions arising from reasoning with the naïve idea of set… russell’s paradox). How do we not know that we won’t revise mathematics again to change our proofs. 1+1=2 might not be true tomorrow!
In philosophy, if we art with an ultimate plurality we have to explain how unity arises. If we start with a unity, we have to explain how plurality arises. I realize this thread is huge. It took me a while to come up with it. I copied people’s ideas throughout the whole post and just put my own twist on them. I can provide references for most of this stuff if you want.
This may seem absurd. It may seem obvious that everyone believes that 1+1=2. However, it might surprise you to learn that not everyone does. Radical monists would state that everything is one. Some versions of Hindu think that plurality is illusion, so on the ultimate level of being, everything is again one. This means 1+1=1.
This shows us that even the simplest arithmetical truths can be sustained in a world-view which acknowledges an ultimate metaphysical plurality in the world. It also must be that there is an ultimate metaphysical unity, since we have “sames.” Two sticks remain sticks even though there are two. In different equations, the number two is the same number, but at different times.
So beginning this thread, we have the problem of the one and the many. We will not talk about that here, though. But to be sure, it is an important problem, and it’ll be raised later.
Some mathematicians do not agree on the validity of certain proofs. For example, intuitionists do not accept the law of excluded middle or the proof by reduction ad absurdum. The human mind is where truth rests. Mathematics is concerned with mental constructions. Therefore proofs that we cannot grasp do not exist. If you aren’t aware of this, and would want an example, ask. Anyhow, this poses a problem.
There are disputes in geometric truth, truths of analysis, and even mathematical existence. I won’t go into this for the sake of a very long thread, but if you want, I can go into it as much as I know.
Anyway, mathematics seems to have at least two problem areas. They are epistemology and metaphysics.
Epistemology
Here we discuss how we come to the knowledge that 1+1=2. Do we know it a priori, or internally, without experience, or do we know it a posteriori, or externally, derived from experience. Plato claims that we gained it by reminiscence or introspection. Bertrand Russell says logical arguments. John Stuart Mill would say from repeated experience. A. J. Ayer says it’s a linguistic convention and not real “knowledge.”
A priori. 1+1=2 is a universal truth. Why should 1 stick plus 1 stick make 2 sticks, though? It might seem ludicrous, but this is a real problem. If we know it, internally, and without experience, why should the external world be contingent and offer us repeated examples of this. If the external world is made up of completely chance matter, why is there a connection? Why don’t sticks disappear and appear randomly while we are counting them? If the outside world has a degree of regularity mixed in with chance elements, why should that regularity coincide with the internal mathematical expectations of the human mind? When you make the Cartesian separation of a priori[/i[ and a posteriori you can’t get them back.
Further, if mathematics is a priori, why do paradoxes arise.. . especially in the form if contradictions… (See Burali-forti’s paradox, or Bertrand Russell’s paradox) or in the form of various counter-intuitive results (Peano’s space-filling curves, Lowenhein-Skolem theorem)… These paradoxes seem to show that saying that mathematics is a priori is not really reliable…
A posteriori. Because of the difficulties of the a priori explanation of mathematics, we have the a posteriori explanation. Mathematical knowledge is inductive. See my induction thread also for better knowledge of induction.. We know that 1+1=2 from repeated examples of one object plus another one making two objects. However, we haven’t seen 234343 objects plus 2343 objects making 236686 objects (I hope my math is correct lol). But we have generalized. But to say this, hints at the a priori answer. Further, why don’t we generalize that 23+1=24 instead of 23+1=25? You can really only say because you have generalized from other generalizations. You do this because you have generalized from previous experience of generalizing, etc…. See my thread on induction. Further, at some point, you might say that this is the way the human mind operates. We then fall into the a priori explanation and it’s problems.
Conventional answer. Mathematicians seem to accept this answer. Mathematics is in some sense a convention of the human language. It isn’t knowledge at all. 1+1=2 because we have agreed to use the words one and two in that way (see Wittgenstein). To say 1+1=2, we are saying A=A in a roundabout way (A.J. Ayer).
These seem like the a priori answer. Why should mathematics prove useful in dealing with the outside world? If it is pure convention… If the conventions are chose because they are useful, we have the a posteriori answer. The conventional answer isn’t really an answer. It’s only a shifting of the question.
Reading Godel’s proof would help you understand the difficulties of all of these answers. I won’t go into it here, because of the massiveness of this thread already.
Metaphysical
The one and the many. 1+1=2 means something. If someone told me that 1+1=2, say a baby for example muttering his first words, I wouldn’t believe that he understood that until he could demonstrate it (do word problems, show the symbols on a chalkboard, etc). So it would be safe to say that we can’t know 1+1=2 until we know many things in relation to that truth. So we are concerned with many experiences and truths in relation to that one.
Modern linguistic theory has pointed out that we can’t understand these symbols without some specification of it’s contrast. 1+1=2 only has meaning as part of a greater system (a plurality). 1+1=2, 1+2=3, and that 1+1=3 is false, etc. We should also understand it on a variational range. 1+1=2, one plus one equals two, I + I = II. We should also understand it distributed in larger units of linguistic behavior and human behavior (proofs using this, word problems, used in calculating your bills, figuring out prices, etc…)
We have to show the unity and stability in this vast amount of truths and experiences though. Some people ask, how can we know anything without knowing everything? This is a valid question. As we have seen, we must know the larger context in which the proof is embedded until we can understand the proof. But we don’t know all of the larger context. How can we? Therefore the next question is how do we not know that the next thing we discover will upset our knowledge.
Physics has changed during the Newtonian revolution, the Einsteinian revolution, and the quantum revolution. Mathematics has revised itself, too, at times (discovering irrationals by the Pythagoreans, discovering contradictions arising from reasoning with the naïve idea of set… russell’s paradox). How do we not know that we won’t revise mathematics again to change our proofs. 1+1=2 might not be true tomorrow!
In philosophy, if we art with an ultimate plurality we have to explain how unity arises. If we start with a unity, we have to explain how plurality arises. I realize this thread is huge. It took me a while to come up with it. I copied people’s ideas throughout the whole post and just put my own twist on them. I can provide references for most of this stuff if you want.
