100% certainty is often Illogical

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First, to get a clear grip on the words we're using here, what do you mean by "apodictic" and "assertoric" evidence? Being a mathematical student myself, I cannot agree with your assertion that no science can formally make statements about truth: mathematics can, to a certain extent, within the limits set by Goedel's Incompleteness Theorems, be used to study its own validity and the truthfulness of its own statements. This all belongs to a field called "metamathematics".

For example, Presburger Arithmetic [PrA] (a particularly simple set of rules for the natural numbers), together with ordinary first-order logic, can be used to prove that Presburger Arithmetic does not contain any contradictions. This is about the strongest proof of validity you can get: the system proves itself flawless. So, in a sense I think PrA is quite "true", if you're willing to posit that the natural numbers are entities about which true and false things can be said.

(Note that Goedel does not apply to PrA, because it is too weak to formulate all of ordinary number theory in. General multiplication, for example, cannot be defined satisfactorally in PrA.)

Of course, you can always say that math isn't a true science. But then what is it?

re apodictic/assertoric: read up on husserl (sounds like a cheap cop-out, but hes far better in explaining than i am)
im not really sure about the terms in english (read most philosophy stuff in german) - but apodictic (evidence) means that its opposite can not be true and the statement itself holds general (as opposed to singular) truth. its that "it isnt possible not to agree"-moment of evidence, whereas assertoric (evidence)means "merely" stating the existence of a singular entity.

i can see your reasoning as a student of mathematics - maths as strict science suffers from exactly what calandale said: its formal conclusiveness comes from axioms (as in every science). maths as acts of thought, however, is a different game. the evidence of 2+2=4 i talk about is the absolute evidence of the mental act, whereas the same statement as a scientific statement would merely derive evidence from axioms.
so, basically, we are talking two completely different things, see?

but, i agree that maths can indeed make statements about the truth of certain ideal entities (and most mathematicians are tactful enough to let it at that). however, these self-proving means sound to me like something that necessarily involves tautology and logic circles, based on the sciences axioms.
the interesting question in regard to truthful statements about said entities is what kind of evidence is providing that truth. i absolutely favor the evidence of the act instead of a system-imanent prove of not being contradictory.

i see i wasnt careful enough in wording my critique of the general opinion on what science is or does: please read "natural science" (is that the correct term in english btw?) instead of "science" - for maths indeed is an exception because of the nature of its subject.
my critique is rather aimed at natural science making statements about truth. the whole method it employs doesnt allow for making such statements, hence i said that it wont allow these for formal reasons. so, sorry for accidentally stepping on your toes this way (i guess i stepped on them a different way with this post )

I can see your point and I respect it, but I think we're talking about very different conceptions of truth here. First, to my mind, mathematics, as being more than just a bunch of empirical observations about geometric figures and about numbers, simply needs axioms and logic (heck, logic is itself based on certain unproved suppositions), otherwise it won't function at all - at least I can't see how it could function otherwise. (But maybe you this see some other way; please tell me.) Your disdain for the axiomatic method seems fair enough from an ontological point of view (do the entities with their properties as described by the axioms and definitions really "exist"?), but much less from an epistemological one, simply because we don't seem to have anything else - except boundless, unconstructive skepticism. Also, your disdain for metamathematics and its techniques ("logical circles") is quite understandable, but in the last essence - I believe - too harsh and self-defeating too. Man as a rational being doesn't have, in any abstract field such as maths, anything but logic to rely on, so it's quite beautiful in my opinion if using logic and axioms you can reach conclusions about logic and the axioms themselves. This is quite different from a circular argument, because I don't suppose what I want to prove. To prove the consistency of first-order logic would indeed be nonsense, but to prove the consistency of for instance Presburger arithmetic is quite sensible. And again, Goedel's famous theorems puts certain restrictions on such efforts, so I think the practitioner of metamathematics is not being arrogant or anything, because he knows the limitations of his own field - in fact, much better than many natural scientists.

Secondly, I think we have to be very clear about the difference between what I call the intrinsic truthfulness or validity of a mathematical system and the truthfulness of the same system in relation to the external world of our perceptions. I call a mathematical system intrinsically valid or truthful or consistent if and only if it is impossible to derive (using standard logic) from the definitions and axioms of the system a contradiction - a statement of the form (A and not-A). So my definition of truthfulness is quite broad - anything goes, as long as it doesn't cause contradictions. But I do recognize that in examining the external world we need totally different criteria - though I'm not very much in the mood to discuss the epistemology and ontology of the natural sciences here. That's such a vast subject, and I would be dead before finishing this post.

One last advice, read the text available at: http://euclid.trentu.ca/math/sb/pcml/pcml.html. Even for the non-mathematician it's quite a relaxed (not *relaxing*) read.

Take care. Bye.

ill look into the text, thanks for the link. (headline looks promising)

i am absolutely with you regarding math being more than empirics. when i refer to the act of thinking, that isnt a reference to empirics (continental philosophy here, not empirism ).
as for axioms: we have to discern two different things here: maths, as an intelligible system, as science, does need axioms for formal reasons. maths as think-acts does not. try thinking without the science stuff about 2+2... you cant but have it make 4 by a forceful moment of evidence (not evidence of a singular happening but evidence of a general "rule" - the same evidence logics have). this evidence doesnt come on behalf of an axiom.

as for truth: i dont build truth with regards to applying or referencing it to the outside world... old st. augustinus even saw that in our perceptions of the world, no truth can be had for certain. descartes kind of worked directly off that, too and ever since hume and (especially) kant, no one would actually deny that.
when looking for truth and according statements, we need to start within ourselves, on the side of the subject, not the object. thats basically why maths can make statements about truth - because its object isnt derived from any possibly fallible source in the supposed world, but is -hierarchically before any experience- already in the subject.

thats basically where i start differing opinions with calandale when he refers to a form of "truer" being (the one idea of the apple as opposed to the pair of apples) beyond ordinary being (which i set as subjective being) - that, to me, is as empty as relying on the subject-constructed outside world of science for unveiling "true" being.

You are misrepresenting the argument. Indeed, I think that
he DOES manage to prove that something which we can
think of nothing greater DOES indeed have existence, of some
sort. But, not that this is the God that one expects. Still, 'tis merely
an argument derived from the words themselves, like presupposing
that there must be something doing the thinking, for thought to occur.



So, we have some system as a model, that we KNOW is incorrect?
How can we expect to find truth with such a model?


The rest of your answers didn't include enough of my
original quote to have any meaning, so I shall refrain from
trying to make the same point again.

No, I disagree entirely. The ontological argument by Anselm is rather ridiculous, it supposes that because God is greater than a thought of him that he must exist, which is nonsensical and little better than the fool and the island. Cogito ergo sum is pretty irrefutable because for thought to occur there must be a thinker as actions cannot exist without something to perform the act.

No, that is a complete distortion of what I said. I merely said that there is no set number of numbers because depending on how things are done it could basically be infinite. The matter of arbitrary divisions is not a matter of correctness or incorrectness because any logically consistent division is correct, but the number of logically consistent divisions is almost limitless.

Using the wrong words damages the argument.
Of course, I also believe that Anselm intended
(but could not state) that the island leads you to
the same thing - a touch of the logos.



Only because thinking is an ACTIVE verb.
See? Again the choice of words is important.
You can't say "thinking happens, therefore I am"
But, by using the phrase "I think..." well you are
already presupposing yourself. At least Anselm
doesn't presuppose God's existence. Hence, 'tis
stronger.



Quite a difference between ALMOST limitless,
and infinite. And that difference is sufficient.
The model is broken.

More irrefutable than Cogito ergo sum??? That is ridiculous, cogito ergo sum is plain and straightforward, Anselm's proof of God is ridiculous, "I imagine a perfect being therefore God exists"?? it almost seems that he proves God by assuming God at the end of his argument rather than maintaining God as an abstract.
[quote]

Descartes' "Cogito ergo sum," was thought irrefutable until Leibniz. Then Hume. Then Kant. Then Hegel, Marx and Nietzsche. Alright, maybe not Leibniz and Marx. They have or had little in common (actually they were no more contemporaries of each other than I am - sorry, not a Time Lord, nor even a companion to one - the Doctor did have SOME male companions over the centuries of his personal time line and the myriad millennia of the history of the Universe and scant decades since 1963) but PERHAPS one common factor was being, while far from irrelevant to the Cogito, not active deconstructors of it. The fact that this may have been because one, the former, did not so much agree with Rene (the fact that this name, the given name of Descartes, does NOT end in a double e reminds you that he was a reborn male unlike those christened - or otherwise initiated into their proper name - Reneé; I hope I did not get that wrong, or at least that no one here knows more French, but the latter is probably a forlorn hope on this site, especially this forum) as go further, while Marx probably did not bother, has been rendered redundant since Derrida (who died quite recently; condolences to any admirers and friends of his who may be reading) opened a gate opening on an abyss with a key loaned by Abbadon while Jacques was attempting to deconstruct the Book of Revelation (revealed nearly a couple of millennia ago, take quite a few decades) in an unpublished manuscript that may not exist - this is getting in a disturbed way. That is bad. If you do not agre, this may be because not considered some of the potential consequences of disturbed frivolity.

Would any grammarian offer to punctuate my paragraphs as a labour of love? I would pay you if I could - do you accept payment in kind, i.e. words? Hopefully more interesting and meaningful ones, though I a poor judge of such things most of the time.

Descartes was a terrible plagiarist.
And a renowned fool.

Thanks for the reply. I find this entire discussion quite enlightening.

Seeing that you refer to mathematics as being, on the one hand, an "act of thought" without the need for axioms, you might want to look into a particular philosophy of mathematics called *intuitionism* (not to be confused with philosophical intuitionism!). It was founded around 1910 by a Dutch mathematician L.E.J. Brouwer - although certain ideas of his go all the way back to Kant. I will not attempt to explain it now, since you may find a much better exposition here: http://plato.stanford.edu/entries/brouwer/. You may find it interesting, if not appealing.

If you have a problem with Bilaniuk's logic text, please PM me (possibly in German, if you like it better). I'll probably be able to help you. Also, you could tell me how much mathematics you know. Especially the technique of proof by mathematical induction is quite essential in understanding mathematical logic and metamathematics.

well, my maths carreer is about having made my a-levels with one of my two focal subjects being maths. my teacher wanted me to study maths, but i initially wanted to study design, to end up studying philosophy now.

if i face unsurmountable problems in the text, ill let you know (am preparing for a test on thursday, so time on off-topic texts is limited). as a student of philosophy, im well aware of the difference of indiction versus deduction (and am very critical of using the latter for ontological statements)

calandale: i wouldnt go as far as calling descartes an idiot, but he certainly is more scholastic than his image - and he indeed wrote very close to st. augustin in some important parts. im a bit sceptical concerning his predecessing role for kants revolution, too... on the one hand, its true that he indeed went farther than his time allowed by starting all of his conclusions from the subject (and the subjects content of thought of the world instead of the world directly), but by upkeeping causality as an ontological principle as opposed to an epistemological one, just didnt get any farther from ontology than his predecessors.
therefore, his cogito ergo sum can be read two ways - as the modern epistemological concept of self-awareness (which indeed is irrefutable) or as strict scholastic argument, which keeps as premiss that in order to act, a certain thing must have "hard" ontological being, thining is evident, hence self is being (whic, in my boat, accidentially is true but by quite messy and speculative reasoning).

edit: if it helps to get the "maths as think-acts" kind of thing: think about it as kants example for synthetic judgement a priori: just focus on what you do while doing that judgement. thats clearly not a statement about maths as science but rather a kind of watch yourself thinking thing.

With all respect, "induction" is used in quite a different (much more specific) sense in mathematics than in philosophy. Proof by induction is a mathematical technique that uses a special but basic property of the positive whole numbers (also called "natural numbers"). Namely: if a proposition P(n), which depends on the natural number variable n, holds true for n = 1, and if the truth of P(N) implies that of P(N+1), then P(n) holds true for *all* natural numbers. Think of it as a domino effect: if I know for certain that the fall of domino n causes the fall of domino n + 1, then I only have to topply domino no. 1 to cause the entire (infinite) chain to collaps. The assumption that P(n) is true, is called the *induction hypothesis*

An easy but non-trivial example: I want to prove that if you add the first n natural numbers, then the result is the half of n times (n + 1); in formula:

1 + 2 + 3 + 4 + ... + n = n(n + 1)/2. (formula 1)

Step 1: Check it for n = 1. This is trivial, since 1 = 1 * (1 + 1)/2.
Step 2: Assume it holds for n = N. This is the induction hypothesis.
Step 3: Prove now, given step 2, that it also holds for n = N + 1.

1 + 2 + 3 + 4 + ... + N + (N + 1) = N(N + 1)/2 + (N + 1) =

(Here, I use the induction hypothesis, by substituting N for n in formula 1).


= N(N + 1)/2 + 2(N + 1)/2

= (N + 1)(N + 2)/2,

but this is exactly formula 1, now with n = N + 1.

So what we've proved now is: if the proposition holds for n = N, then it's also true for n = N + 1.

Step 4: By mathematical induction we're done now. The fact asserted in formula 1 is true for every natural number.


The best of luck with your test!

thankies for the luck. im glad itll be done soon... i cant see avicenna and descartes anymore. -_- (i so wanted to have schopenhauer or something fun as test subject...)

i see about induction. over at you guys, it means a method, whereas we use it as opposed to deduction. induction being to derive the specific from the general and deductive meaning to deduce the general from the specific.