Mathematical Epistemology

[Kyrisch]

Often I speak about how it is impossible ever to know anything as a true and unassailable fact, that assumptions are required in all things (even the scientific method, which attempts to be as objective as possible, operates on the unknowable assumption that observations of the material world can inform of its workings).

 

Invariably, I am rejoined with: "But we know 2+2=4! That is irrefutable and unassailable, and requires no assumptions."

 

This disturbed me, because it seemed to violate my previously stated ideology, so I thought deeply on it, and I came to the conclusion that mathematical statements like that are actually tautologies -- that four is defined as the sum of two and two, and that the idea of four -- the actual, abstract idea, consists in and of itself of a sum of two and two.

 

Further, in another thread (one about perpetual motion), a cracked pot declared that "anything is possible if you believe that it is", to which (I believe it was iNow) replied "Unfortunately, there is no way to find a rational square root of two, no matter if you believe contrariwise".

 

Extending the conceptual basis I just introduced, this additional mathematical "truth" can just as well be reduced to tautology, however complex, in that the square root of two is just the number which, multiplied by itself results in two, which hearkens back to the statement 2+2=4. But that the square root of two squared equals two is not being questioned here, rather its identity as rational. However, rationality is again defined to be the quality of ability of representation by the quotient of two integers.

 

In conclusion, it seems to me that the whole body of math does not include any actual knowledge, just an infrastructure of definition based entirely on the abstract idea of quantity.

[BigMoosie]

Invariably, I am rejoined with: "But we know 2+2=4! That is irrefutable and unassailable, and requires no assumptions."

 

This disturbed me, because it seemed to violate my previously stated ideology, so I thought deeply on it, and I came to the conclusion that mathematical statements like that are actually tautologies -- that four is defined as the sum of two and two, and that the idea of four -- the actual, abstract idea, consists in and of itself of a sum of two and two.

 

The eqality [math]2 + 2 = 4[/math] does require an assumption:

 

* For every natural number n, Successor(n) is a natural number.

 

See the wikipedia entry on Peano axioms.

 

In conclusion, it seems to me that the whole body of math does not include any actual knowledge, just an infrastructure of definition based entirely on the abstract idea of quantity.

 

I agree, but I wouldn't say it doesn't contain any actual knowledge.

[Sisyphus]

What's a circle?

 

You're not entirely wrong - pretty much anything you encounter in mathematics can be deduced from a handful of axioms and definitions. See Euclid's Elements for the prototypical example of this. It's more than just a notion of quantity, though.

 

It's also not just definition. You can define 4 as 2+2, but then you still have to determine whether it is equal to 3+1. You can define it as "that which is equal to 2+2, 3+1, 5-1, etc.," but that's an infinite definition and therefore incomprehensible, and it still hasn't been determined that such a thing is self-consistent. You can define a "blerg" as something which is equal to both 2 and 3; is that just as meaningful as the statement 2+2=4?

 

More fundamentally than that, though, is the continued assumption that deduction 100% free of error is possible, i.e. that reason itself is trustworthy. It seems obvious that 2+2=4, and it's something we all agree on. But are we all mistaken? Do we, in fact, all agree, or are both our memories concerning it flawed in the same way? Do I even agree with you, or are we misunderstanding one another? Am I even real?

Edited April 27, 2009 by Sisyphus

[Kyrisch]

The eqality [math]2 + 2 = 4[/math] does require an assumption:

 

* For every natural number n, Successor(n) is a natural number.

 

But Peano's Axioms themselves come solely as a natural consequence of the idea of quantity. Such properties as reflexive equality, transitive equality, and symmetric equality are necessary and logical consequences required to maintain coherency in mathematical language. As such, these fall into the broad heading of everything that has been built upon the concept of quantity and not, essentially, assumptions. They are more definitions denoted by mathematicians than core assumptions taken as truth.

 

I agree, but I wouldn't say it doesn't contain any actual knowledge.

 

Then of what would you say the infrastructure of mathematics consists?

 

What's a circle?

Not sure how this relates, but it is the curve containing all points equidistant from a single point in two-space.

 

You're not entirely wrong - pretty much anything you encounter in mathematics can be deduced from a handful of axioms and definitions.

 

I'm rather arguing that even the core axioms and definitions are themselves naturally and logically consequential and are merely tautologies of the concept of quantity.

 

It's also not just definition. You can define 4 as 2+2, but then you still have to determine whether it is equal to 3+1. You can define it as "that which is equal to 2+2, 3+1, 5-1, etc.," but that's an infinite definition and therefore incomprehensible, and it still hasn't been determined that such a thing is self-consistent. You can define a "blerg" as something which is equal to both 2 and 3; is that just as meaningful as the statement 2+2=4?

 

You misunderstood me again. I am not saying that we simply 'named' the sum of 2 and 2 "four" or even denoted this quantity with the symbol "four", but rather that the very idea of the quantity FOUR includes all of those things in its abstract conception.

 

For instance, if we have four blocks, it's abstractly equivalent to having two blocks and two more, five blocks and one anti-block, three blocks and one more. A person conceptualizing four is like viewing four blocks -- all of these things are obviously represented just by the mere conceptualization of the quantity.

 

More fundamentally than that, though, is the continued assumption that deduction 100% free of error is possible, i.e. that reason itself is trustworthy. It seems obvious that 2+2=4, and it's something we all agree on. But are we all mistaken? Do we, in fact, all agree, or are both our memories concerning it flawed in the same way? Do I even agree with you, or are we misunderstanding one another? Am I even real?

 

This bit is actually in line with my introduction in the OP. I hold the philosophy that there is no such thing as proof of anything, and that we cannot actually know that we know what we think we know. I admit I'm working under the assumption that logic works, but you must remember that it is an assumption (that goes along with the assumption that observation can inform us about reality) on the list of those assumptions without which we cannot live in sanity.

[ajb]

If we are talking about mathematics in general, then anything is only as true as it is consistent with your original axioms/definitions (and maybe even on how you apply logic and proof theory, see topos theory. You can have set-ups that don't allow proof by induction (for example)!).

 

I would say that mathematics does contain knowledge rather than just tautologies as many theorems/statements/lemmas etc can be highly non-trivial and not obvious from the initial set-up. However, to an extent I agree with the idea that a statement is "trivial" once it has been proved. It is a fact and will remain a fact, up to the original constructions.

 

 

What I think is confusing the issue here is that statements like "2+2=4" is that it can be "experimentally proved". That is we can take 2 apples, then take another 2 apples and clearly we have 4 apples.

 

But mathematics deals with things that are much more general than this and do not (for sure not directly) rely on the idea of a quantity. Despite this is seems that mathematics does tell us a lot about nature. Maybe this is where the true knowledge lies.

[Kyrisch]

I would say that mathematics does contain knowledge rather than just tautologies as many theorems/statements/lemmas etc can be highly non-trivial and not obvious from the initial set-up.

 

I'm curious as to which theorems/statements/lemmas you had in mind when you said this?

[Sisyphus]

Not sure how this relates, but it is the curve containing all points equidistant from a single point in two-space.

 

It relates because it is a simple example of a mathematical concept that is not merely contained in the definition of quantity. There is more that you need, even if those things seem "so obvious they don't need saying" to the average human, and such it wouldn't occur to you that they are in fact necessay axioms.

 

You misunderstood me again. I am not saying that we simply 'named' the sum of 2 and 2 "four" or even denoted this quantity with the symbol "four", but rather that the very idea of the quantity FOUR includes all of those things in its abstract conception.

 

For instance, if we have four blocks, it's abstractly equivalent to having two blocks and two more, five blocks and one anti-block, three blocks and one more. A person conceptualizing four is like viewing four blocks -- all of these things are obviously represented just by the mere conceptualization of the quantity.

 

That 5-1=4 does indeed follow necessarily, a priori, from any other definition of 4 and the conventional rules of arithmetic. That doesn't mean it's tautological, however, any more or less than the conclusion of a valid syllogism is a tautology of its premises. Whether you want to say it's "contained in" or "necessarily follows from" the others is pretty much just semantic, but it is saying something additional. It needs to be deduced in accordance with rules, and the fact that 2+2=4 and 5-1=4 requires justification. (That justification being the difference between 5-1=4 and 6-1=4.)

 

I'd also be careful with "mere conceptualization." There are an infinite number of arithmetic operations that yield an answer of four, so clearly nobody has ever conceptualized everything "four" implies, yet that doesn't mean nobody has ever said the word "four" with any meaning.

 

This bit is actually in line with my introduction in the OP. I hold the philosophy that there is no such thing as proof of anything, and that we cannot actually know that we know what we think we know. I admit I'm working under the assumption that logic works, but you must remember that it is an assumption (that goes along with the assumption that observation can inform us about reality) on the list of those assumptions without which we cannot live in sanity.

 

That's all true, for the most part. But again, there's a difference between accepting certain axioms because you don't really have a choice, and in declaring that they are not axioms. Sometimes, in fact, they turn out not to be necessary. For thousands of years the idea of infinitely equidistant parallel lines was "contained in" the idea of right angles for all intents and purposes, but as an identified axiom, it was able to be declared false, and it turned out that a consistent geometry could be constructed nonetheless.

Edited April 28, 2009 by Sisyphus

[ajb]

I'm curious as to which theorems/statements/lemmas you had in mind when you said this?

 

Nothing in particular.

 

Everything looks trivial once you have a proof.

 

What about the classification of finite simple groups?

 

Tens of thousands of pages of work. It is claimed that no one individual understands all of it and there maybe errors along the way.

[Kyrisch]

It relates because it is a simple example of a mathematical concept that is not merely contained in the definition of quantity. There is more that you need, even if those things seem "so obvious they don't need saying" to the average human, and such it wouldn't occur to you that they are in fact necessay axioms.

 

It does not take anything else but the comprehension of the concept of quantity to understand the definition of a circle. (Well, besides what a 'point' is, but that's the core geometric concept, and this thread is rather more about numbers in general, and the relationship between numbers and operations).

 

That 5-1=4 does indeed follow necessarily, a priori, from any other definition of 4 and the conventional rules of arithmetic. That doesn't mean it's tautological, however, any more or less than the conclusion of a valid syllogism is a tautology of its premises.

 

A priori knowledge or justification is independent of experience (for example 'All bachelors are unmarried') from wikipedia. The example it gives is a tautology, that bachelors are defined as unmarried. I think the two concepts are intimately related.

 

I'd also be careful with "mere conceptualization." There are an infinite number of arithmetic operations that yield an answer of four, so clearly nobody has ever conceptualized everything "four" implies, yet that doesn't mean nobody has ever said the word "four" with any meaning.

 

Well no, but I am talking about mathematical knowledge as a body outside of the human experience that we are "discovering". And I never said that "four" has no meaning, but rather that it has ALL of the meaning contained therein.

[Pangloss]

Moved to Pseudoscience and Speculations.

[Kyrisch]

Moved to Pseudoscience and Speculations.

 

May I ask why?

[ajb]

May I ask why?

 

 

I would like to second that.

 

I thought that this thread was much to do about mathematical philosophy and foundational issues. I agree it may not exactly be "mathematics" but is is about mathematics.

[Royston]

I thought that this thread was much to do about mathematical philosophy and foundational issues.

 

...and a lot better than similar discussions that have been left in the maths forums. So I third the notion. (sorry Pangloss.)

[iNow]

So, let's move this back to an appropriate forum and continue with the discussion. It's obvious to the community that P&S is not its proper home.

 

Once it's moved, you can also delete all of these OT posts asking why it got moved to ensure continuity...

[Klaynos]

Logica's posts and replies have been moved to Logica's thoughts on numbers for their own discussion.

 

Moved back to maths.