Oh one last point…calculus, while composed of “theorems” indeed is used to posit “theories”…that was the whole reason Newton invented inferential and differential calculus. Perhaps whoever said differently (warren??) wasn’t paying attention in Physics class.
What?
Every other science is a human creation. Mathematicians uncover the laws of the universe that existed long before man.
Oversimplified for you reading pleasure.
Epistemology is the philosophical inquiry into the nature of knowlege. How do we know what we know? How do we know that we know what we think we know? As you can see, it’s complicated just to pose the question, much less answer it.
In Mathematical Epistemology, we are exploring propositions (theorems) deducible from a set of axioms. A specific proposition is true or false according to a particular set of axioms. Propositions are not true or false “on their own merit” so to speak. The proposition that 1+1=2 is true according to the axioms of arithmetic.
An axiom set is either consistent or inconsistent. If the axiom set is consistent, then it is impossible to correctly derive a proposition that is the inverse of another correctly derived proposition; i.e. p and not p are not both true theorems of the axiom set.
Godel proved that it is impossible to use theorems of an axiom set to prove that the selfsame axiom set is consistent. Therefore, to prove an axiom set consistent, we have to create a new axiom set according to which we can prove the first consistent. But we now have an infinite regress; we cannot use the second axiom set to prove itself, and must create a third to prove the second against, ad infinitum.
We also have the “meta-mathematics” of propositional calculus which formalizes the ordinary rules of deduction; for instance “The expression p or q is true if and only if one or both of p and q are true.”
However, being an axiom set, Godel’s theorem applies to propositional calculus as well! One cannot use propositional calculus to prove its own consistency. So not only are arbitary axiom sets not provably consistent, the usual method of deduction is also not provably consistent. Needless to say, these conclusions give mathematitians, who consider themselves seekers after absolute truth, a queasy feeling, and most of them don’t like to discuss the situation.
Scientific Epistemology is somewhat more forgiving. The tests of scientific knowlege depend on predictive power (e.g. physics) or narrative consistency (e.g. cosmology). A scientific theory, an explanation of the nature and behavior of physical phenomena, is considered “strong” if it makes consistently true preditions about directly observable phemonoma. It is considered “robust” if it makes correct predictions under a wide range of circumstances.
The Scientific Method operates much like Sherlock Holmes’ dictum: “When you have eliminated the impossible, whatever remains, however improbable, is the truth.” They seek to create a falsifiable hypothesis to explain a phenomenon, and then try their darndest to prove that hypothesis false.
Scientific Epistemology also has its fundamental difficulties. The most obvious being the “hypothesis creation” problem, that Pirsig eloquently describes in Zen and the Art of Motorcycle Maintenance. There are an infinity of hypotheses that could explain a given phenomenon; how do we choose which to test according to the Scientific method?
And then there’s Quantum Mechanics. QM is most definitely a very strong and very robust theory according the basic definition of Scientific Epistemology. The fact that you can read this message at all on your computer demonstrates that every second, billions of predictions from QM are proving themselves true. The problem is that features of ordinary reality we consider fundamental, e.g. definiteness, locality, causality, do not apply to the quantum microworld. Even time itself can appear as an ordinary spacial dimension in QM. With the Copenhagen Interpretation of QM, scientists finally in effect threw up their hands and said, “we have no idea what is really going on in the microworld; QM is just a mathematical fiction to predict the results of experiments.”
It also turns out, apparently coincidentally, that almost every interesting pure mathematical system (axiom set + theorems) turns out to be a powerful way of generating theories with strong and robust predictive power. It is very unsettling to see this almost perfect correspondence between abstract creations of the human mind, and the complicated behavior of seemingly unrelated physical phenomena.
The human mind has evolved to both seek and create patterns from empirical phenomena. It doesn’t want to stop even when confronted with the limitations of ordinary knowlege. Ultimately the choice between a scientific or religious interpretation of the limits of knowlege is an aethetic choice. The religious interpretation sees the fundamental nature of reality as transcendent and purposeful; the scientific sees it as purposeless and possibly perverse. It seems that not only does God roll dice, but sometimes he rolls them where we can’t see them!
To reply to Lance Turbo who’s faith in mathematics and science is unshakeable…mathematics has nothing to do with universal princples in a direct sense…mathematics does not exist in the “real world” It is merely a tool invented by humans to attempt to help them understand a very complex universe…much like a telescope helps us see distant things, mathematics helps us understand complex physical interactions, but both are human creations.
throughout history, attempts to reconcile mathematics and reality have generally been subsequently proven wrong…thus the forward march of science…the mathematical equations used to demonstrate physical and astrophysical phenomenon employed by Ptolemy, Newton, Kepler, and Einstein in turn have all been demonstrated to have errors. Even our two current leading theories in physics…Relativity and Quantum Physics, contradict each other mathematically. They can’t both be right, and probably neither are.
Thus mathematics is a human abstraction not a “real” phenomenon.
To reply to Lance Turbo who’s faith in mathematics and science is unshakeable…mathematics has nothing to do with universal princples in a direct sense…mathematics does not exist in the “real world” It is merely a tool invented by humans to attempt to help them understand a very complex universe…much like a telescope helps us see distant things, mathematics helps us understand complex physical interactions, but both are human creations.
throughout history, attempts to reconcile mathematics and reality have generally been subsequently proven wrong…thus the forward march of science…the mathematical equations used to demonstrate physical and astrophysical phenomenon employed by Ptolemy, Newton, Kepler, and Einstein in turn have all been demonstrated to have errors. Even our two current leading theories in physics…Relativity and Quantum Physics, contradict each other mathematically. They can’t both be right, and probably neither are.
Thus mathematics is a human abstraction not a “real”
phenomenon. And yes every other science is a human creation as well…and equally subject to the inconsistencies, fallacies and flights of fantasy as mathematics.
To reply to Lance Turbo who’s faith in mathematics and science is unshakeable…mathematics has nothing to do with universal princples in a direct sense…mathematics does not exist in the “real world” It is merely a tool invented by humans to attempt to help them understand a very complex universe…much like a telescope helps us see distant things, mathematics helps us understand complex physical interactions, but both are human creations.
throughout history, attempts to reconcile mathematics and reality have generally been subsequently proven wrong…thus the forward march of science…the mathematical equations used to demonstrate physical and astrophysical phenomenon employed by Ptolemy, Newton, Kepler, and Einstein in turn have all been demonstrated to have errors. Even our two current leading theories in physics…Relativity and Quantum Physics, contradict each other mathematically. They can’t both be right, and probably neither are.
Thus mathematics is a human abstraction not a “real”
phenomenon. And yes every other science is a human creation as well…and equally subject to the inconsistencies, fallacies and flights of fantasy as mathematics.
avalongod wrote:
I was paying attention in Physics but you weren’t paying attention in English. When I wrote ‘sports’, I meant ‘sports’. I didn’t mipsled ‘supports’ and I was referring only to calculus.
Your rather obvious point that physical sciences use mathematics to explain observed phenomena and have invented many branches of mathematics to assist themselves is not news to me. I also know that there are many fields of mathematics that have laid fallow of any physical application before being applied.
Andrew Warinner
To reply to Lance Turbo who’s faith in mathematics and science is unshakeable…mathematics has nothing to do with universal princples in a direct sense…mathematics does not exist in the “real world” It is merely a tool invented by humans to attempt to help them understand a very complex universe…much like a telescope helps us see distant things, mathematics helps us understand complex physical interactions, but both are human creations.
throughout history, attempts to reconcile mathematics and reality have generally been subsequently proven wrong…thus the forward march of science…the mathematical equations used to demonstrate physical and astrophysical phenomenon employed by Ptolemy, Newton, Kepler, and Einstein in turn have all been demonstrated to have errors. Even our two current leading theories in physics…Relativity and Quantum Physics, contradict each other mathematically. They can’t both be right, and probably neither are.
Thus mathematics is a human abstraction not a “real”
phenomenon. And yes every other science is a human creation as well…and equally subject to the inconsistencies, fallacies and flights of fantasy as mathematics.
avalongod wrote:
I was paying attention in Physics but you weren’t paying attention in English. When I wrote ‘sports’, I meant ‘sports’. I didn’t mipsled ‘supports’ and I was referring only to calculus.
Your rather obvious point that physical sciences use mathematics to explain observed phenomena and have invented many branches of mathematics to assist themselves is not news to me. I also know that there are many fields of mathematics that have laid fallow of any physical application before being applied.
Andrew Warinner
avalongod wrote:
I was paying attention in Physics but you weren’t paying attention in English. When I wrote ‘sports’, I meant ‘sports’. I didn’t mipsled ‘supports’ and I was referring only to calculus.
Your rather obvious point that physical sciences use mathematics to explain observed phenomena and have invented many branches of mathematics to assist themselves is not news to me. I also know that there are many fields of mathematics that have laid fallow of any physical application before being applied.
Andrew Warinner
If it wasn’t new to you, why did you suggest otherwise earlier. Perhaps you’re superior logic and intellect elude me (but I doubt it)
By the way…“Mipsled”?? Cute.
avalongod said:
[quote]
calculus, while composed of “theorems” indeed is
used to posit “theories”…that was the whole reason
Newton invented inferential and differential calculus.
[quote]
My dictionary defines posit as ‘to assume or affirm the
existence of’. Calculus never ‘posited’ anything,
certainly nothing concerning the reality of physical
law. Newton devised/used calculus to quantify the rate
of instantaneous change. Calculus provides a means of,
for example, calculating the nature of planetary orbits and and the roots of equations. Calculus does not require
that the orbits be elliptical or the roots be rational.
Also are you sure you mean ‘inferential’?
Good point and I stand corrected.
I seem to remember a notation on Newton as having invented an “infential” calculus, though perhaps that fell out of use.
avalongod wrote:
I wouldn’t claim to be logic personified as you suggest, but where did I say that where the development of mathematics wasn’t often driven by practical application?
I objected to the analogy between physics, where some theories, say Aristolean mechanics, is proved ‘wrong’ by the introduction of new theory, Newtonian mechanics, and mathematics, where no rigorously proved conjecture is ‘wrong’ if one accepts the logical assumptions on which it is built.
Physical sciences and mathematics differ in another important respect. Physical theories are not so much disproven as supplanted in some circumstances. Einstein’s General Theory of Relativity did not really disprove Newtonian mechanics as both are still equally predictive in certain conditions.
By the way, Newton invented fluxional calculus, while Leibnitz invented differential calculus. The difference between the two is largely notational - though modern calculus mostly uses Leibnitz’s notation and methods. Neither ‘proved’ their mathematical work to the standards of modern mathematical rigor. Indeed, much of the history of 19th century mathematics was devoted to cleaning up their mess which in turn inspired much of the attention to mathematical rigor in the late 19th and early 20th centuries.
Andrew Warinner
Trust your memory but verify: it’s Gottfried Wilhelm von Leibniz, no ‘t’.
Andrew Warinner
For all those who says that math can be disprooven because it is a man made system, ask yourself wheter or not present existence of the sun can be disprooven just because we named that ‘thing’ a sun. Does it make sun a ‘human invention’?
2+2!=5 and in decimal system will never be equal to 5, and if you will take any other system it will be still in the very concept 4 ( whenever it will be 101 in binary or yellow in color axed system ). Even when you would create an inconsistenced axis where there would be nothing between 4 and 7 ( assuming that 4 is just 3 times the size of 1 for some reason ) it would be 4.(3) which would be still in the very same concept 4.
If this is to much for humanist-leftist brain go to the pool house, put two balls at the left part of the table and another two balls on the right part. Now combine them and count how many you get. I dont know what bizzare world it had to be, to just by combining two sets of pysychaclly and chemically ‘unactive’ objects creating another one acompanied with sparks poofs and fireballs.
Regards
Ziom
Hi, trunks200, welcome to the SDMB.
First of all, you’ve revived a conversation from the year 2000, and many of its participants are no longer still around (though some of them are).
Now, to the essence of your comment: there are different schools of the philosophy of mathematics—basically, different beliefs about what math essentially is (Formalism, Intuitionism, Logicism, Platonism, etc.) Your claim that mathematics is about something that has a real existence outside of the people doing the math is one that some, but not all, of these schools of thought would agree with.
I don’t understand them well enough myself to give a good, succinct summary of the different points of view, but you (or anyone else who’s interested) might look at one of these links for more details (the first of which seems to most directly address what you are saying):
https://www.dpmms.cam.ac.uk/~wtg10/philosophy.html
http://plato.stanford.edu/entries/philosophy-mathematics/
Thudlow Boink
Yes but the topic is still open which means any new message will stack it up and in the end ressurect it.
There are things that are indisputable and this one is exactly like that.
No matter how many opinions may be on the topic that true!=false only one can be right.
Disscussion can be made on wheter or not some object possess some attribute but tere can not be a disscussion about attribute being an attribute.
Regards
Ziom
If something has been disputed, then it is not indisputable. QED. 
Here is something that is indisputable. Now either i’m wrong and there are no indisputable things, or you are wrong and there are ones. Either way we leave with one indisputed conclussion.
Regards
Ziom