Intuitionism vs Formalism. Which is better?

.99 = 1?

1 + 1

lol

48/2(9+3) = ?

formalism

288

or 2?

Intuition isn't math?!?

As in approach to mathematical theoretical development?

Both Bleh.

I'm not a researcher in the maths area :P

What is the actual answer to that 288 vs. 2 thing? I was always trained to distribute the 2 across the parentheses before dividing, which would mean its 2.

isn't that simply order of operations that are taught in year 6 primary school?

EDIT: at least in Australian primary schools

order of operations says the answer is 288, 48/2 = 24 * (9+3) = 288, but that's mostly because the question is written too ambiguously
I get the answer of 2, due to something called "implied multiplication", which is a supposedly debated difference to the regular order of operations. It suggests that any number directly before parentheses takes priority over regular muliplication/division (which makes sense imo)
anyway, to get a definitive answer, it would need to be written differently... either (48/2)(9+3) or 48/(2(9+3))

Stop hijacking my thread!!

Slagpit - why formalism? How do they satisfy the intuitionist's objection to the axiom of choice? Why is the law of the excluded middle acceptable?

Could you provide me better references than wikipedia?

Here are the aspects I have keyed in on from wikipedia:

"According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other contensive subject matter — in fact, they aren't "about" anything at all. They are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation (or semantics)."

"In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied. Instead, logic and mathematics are the application of internally consistent methods to realize more complex mental constructs."

Clearly I can not discuss any subtleties as you may wish but barring various subtleties those two statements seem very similar to me.

Formalism believes that the language creates the truths, basically. You create a system of symbols and rules of logic, and from those rules of logic and the definitions of all of your symbols, you create a mathematical system. Peano arithmetic, euclidean geometry, riemannian geometry, etc are all different mathematical systems, generated by a finite set of axioms that can all be written in pure mathematical language.

Intuitionism believes that truths are limited to those which can be proven to be true. They believe that existence statements are not valid, that one cannot simply state an object exists unless one demonstrates how to find or create that object. Therefore, they reject some of the truths that formalists create with their mathematical language.

The law of the excluded middle is a touchy point for mathematicians. The law states that every proposition is either true or false. Mathematicians rely heavily on this principle, especially formalists. In logical language, the law of the excluded middle states that the proposition "p or not p" is true. Intuitionists object to this claim because "p or not p" requires that either 'p' is true, or 'not p' is true. They believe that one should be required to demonstrate that 'p' is true, or that 'not p is true' as a proof of 'p or not p' is true.

It is extremely inconvenient to have to actually proof a statement or its negation. A simple example of a statement which has not yet been proven, not its negation been proven is the Goldbach conjecture, that every even number greater than two is the sum of two prime numbers.

So far mathematicians have not proven the Goldbach conjecture to be true, nor found a counterexample to prove the Goldbach conjecture false.

Godel's incompleteness theorems actually prove the existence of true but unprovable theorems in mathematical systems constructed a certain way. Godel proved the impossibility of proving the consistency of arithmetic within formalist language. His incompleteness theorems point towards the necessity of the strict requirements of intuitionism, in my opinion.

The inconveniences of intuitionism tend to override its adherence to truth for most mathematicians, however. They seem content to use "not proven false" as equivalent to "proven true", which I object to.




edit: and since I am an intuitionist, my analysis of the intuitionist vs formalism debate is biased. I'm hoping to encounter a formalist on here to debate, maybe slagpit?

mathematics is definately a philosophy rather than a science :P

I would argue that since all mathematics boils down to set theory which is a way of representing logic. Logic is a man made set of rules and there can be different kinds logics. Hence I would promote formalism.

I conceded that you can actually "prove" that certain things in mathematics cannot be "proven" (for example there are an infinite number of "infinities" but one can not show what type of "infinite" that is). This is a consequence of the way we've defined number theory.

Just because you cannot "prove something to be true" does not make it false imo since truth is subjective. A good example of this is the famous goat question in statistics where the answer depends on the viewers perception of reality. Another example (from science) is relativity in a sense.

I'm really tired right now so I'm not sure my argument is completely clear or coherent but I would like to continue this debate when I can think straight.

Clearly the awnser is 42.

So do you accept the law of the excluded middle? Or do you reject it?

The point of intuitionists is that just because something is 'not false' does not necessarily make it true. Intuitionists do not claim that the inability to prove a proposition makes it false, they merely claim that it makes it unproven. Therefore, a proposition is not either true or false, like formalists like to assert. To an intuitionist, a proposition is true, false, or unproven because intuitionists do not accept the law of the excluded middle.

Therefore, to an intuitionist, "p or not p" is not a given.

How can one argue that "p or not p" is not a given?

It seems to me that "p or not p" is an exhaustive set? We may not be able to determine which it is, but we should be able to be certain that it falls into that set somewhere...

See the example of the Goldbach Conjecture about even numbers.

Or for a different type of example, see the Continuum Hypothesis in ZF and ZFC set theory. It has been proven by Godel & Cohen that the Continuum Hypothesis cannot be proven or disproven in ZF and ZFC set theory. So one could say that the Continuum Hypothesis contradicts 'p or not p'.

And if you're really crazy good at mathematics, take a look at Godel's incompleteness theorems and try to make sense of those.

the answer is 17

I should add "Just because you cannot "prove something to be true" does not make it false" and just because you cannot prove something to be false does not make it true (forgot to add that).
So I would argue that a statement is indeed true/false/ or unproven. But then my area of mathematical knowledge is primarily statistics. I can cite examples from there but I better hold off until I can actually think straight.

Would a better example be the statement Team A will win tonight. Currently we can neither prove nor disprove the statement, even though it is either true or false. Similarly, given the axioms we have accepted it is indeterminable whether a statement is true or false. BTW Rockman, I don't think anyone thinks the Goldbach Conjecture is not decidable just that is hasn't yet been proven or disproven...

Your last sentence is rather imprecise. By 'think' do you mean that they consider it more likely than not that the Goldbach Conjecture is decidable? Or do you mean that they think it to be highly unlikely that it is undecidable and would register genuine surprise if it was proven to be undecidable, the same way that the mathematics community reacted to Godel's proofs and their consequences?

this: "they think it to be highly unlikely that it is undecidable and would register genuine surprise if it was proven to be undecidable, the same way that the mathematics community reacted to Godel's proofs and their consequences?"

I would have to disagree with you then. Godel proved that there are true statements in arithmetic which cannot be proven or disproven, and I would not be terribly surprised if the Goldbach Conjecture was proven to be independent of the axioms of arithmetic.

Due to the ability of mathematicians in recent years to prove the independence of statements from various axiom systems (such as CH being independent of ZFC), I think that the Goldbach Conjecture may end up having the same result.

And I don't think that I am alone in this belief. I do not expect a proof of its independence during my lifetime, but I think that many many mathematicians would, like me, not be surprised if such a proof was given.

.99 =/= 1

.99999... = 1

That statement will eventually be proven or disproven. Instead choose a statement that has a possibility of never being proven or disproven.

Inside finitist systems, intuitionists have no problem with the law of the excluded middle. They object to the use of the law of the excluded middle when dealing with infinite systems. Unlike finitists like Kroenecker, they don't have a problem with infinite systems themselves. They do have a problem with non-constructivist infinite systems, i.e. systems whose cardinality is not omega zero. But inside infinite systems with a cardinality equal to omega zero, they have a problem with the law of the excluded middle, but not with the claim that certain truths in the system can be adequately proven.

I should qualify that in no way is this close to any of the math I do, but...

If the Goldbach conjecture were false that would be provable (give a number not the sum of two primes). So if the conjecture were proven undecidable, then that would actually be a proof of its truth ...

But again, this is just an impression, could very well be wrong, but apart from logicians, I think most people think that the conjecture is true and a proof exists but has not yet been formulated...

That wouldn't be a proof of its truth. It would be a proof that it is undecidable.

If it was true, then assuming that there exists an even number (other than two) which is not the sum of two prime numbers would eventually lead to a contradiction.

If it is proven undecidable, then assuming that there exists an even number (other than two) which is not the sum of two prime numbers would never lead to a contradiction.

Proving it to be true means that if you assume it to be false, you will eventually end up with a contradiction.

That is the difference between proven undecidable and proven true.

I don't think I'm qualified to have such a debate. It was also my understanding that there are more philosophies relating to mathematics than the two you listed, so simply preferring formalism over intuitionism would not imply that I'm a formalist?

I'm not sure I fully understand what your saying, but undecidable is with respect an underlying set of axioms (what I was trying to analogize with my statement about Team A earlier, we can't prove or disprove because of a lack of time, (equiv lack of axioms)).
Anyways, given the axioms of Peano aritmetic if a statement is undecidable then their exist sets of "numbers" satisfying the axioms such that there exist even numbers which are not the sum of two primes, and that there exist sets of "numbers" satisfying the axioms such that any even number is the sum of two primes. But because any system of "numbers" satisfying the axioms of P. aritmetic must include the set of numbers 1,2,... that we usually refer to, then necessarily the statement is true for this set of numbers....

Anyways, I gotta run out soon and will be gone for the weekend. But its always good to talk Rockman, like I said this isn't something I know much (or possibly anything about), so it could be nothing I'm saying makes any sense. Peace=)

There are many more. Many of the schools are in varying degrees of agreement with each other. Intuitionism and Formalism are probably the two most well known schools of mathematical philosophy, and are also in strong disagreement with each other.

Formalism is mainly attributed to David Hilbert. Intuitionism is mainly attributed to LEJ Brouwer. Mathematical platonism is the description for Kurt Godel's views. Finitism is the description for Kroenecker's views. Constructivism is very closely related to intuitionism. Mathematical realism is very closely related to mathematical platonism. The list goes on and on as people create new names for their slightly different mathematical philosophies.

The book I am reading right now breaks it down into three different philosophies. Formalism from Hilbert's view, Intuitionism from Brouwer's view, and Logicism from Frege's view. Logicism is kind of a middle ground between the two, which is why I asked about Formalism versus Intuitionism without including logicism.

I think I get what you're saying, and if I understand you properly, then you are correct.

If the goldbach conjecture is undecidable, then...

The system of peano arithmetic + the goldbach conjecture would be consistent.
Additionally, the system of peano arithmetic + the negation of the goldbach conjecture would also be consistent.

Further complicating things is Godel's proof, which indicates that the system of peano arithmetic can not prove its own consistency, and only a metamathematical proof could prove the consistency of peano arithmetic.

Why an either/or,

why not a Both/And,

Understanding that both approaches have strengths and weaknesses that can be brought to the table, and that while on occasion they may result in differences, oftentimes looking at things through both lenses can bring a depth of clarity not avialable through looking only at one lense.

Is your Omega zero the same as Aleph naught?

I also feel that it doesn't have to be an either or, but as a physicist I tend to lean towards formalism. I would agree that truth is limited to what is proven true, but that given some axioms or statement you can then deduce a result, which could be useful as a physical interpretation.

More recently, I would say that some of the topological modular forms based on euclidean field theories with a super-algebra is such a case, that given some axioms of properties you would imagine that they would have, you can get a nice relationship between them and the Witten genus of N=1/2 supersymmetry which could be useful though not necessarily proven to be true. There are quite a few more prominent examples of this as well, but maybe that's different from what Rockman is arguing against.

*this proves just one thing...you are all nerds. Congrats(5)

I need an asprin....or a beer

Yes, they are the same. In college, I got in the habit of using omega sub-zero and omega sub-one rather than aleph naught and aleph one, because aleph looked too much like the N used for the set of Natural Numbers.

You summed up the formalist view pretty well. Its about being useful. The law of the excluded middle is used because it is a very useful law. It is also a law which seems intuitive, which is kinda funny because it is a law that the intuitionists reject.

If theory can not be disproven then it is not a good theory.