'What was the greatest discovery in history?'

[a_unique_person]

According to this (http://www.abc.net.au/rn/scienceshow/stories/2006/1807626.htm#transcript), Kurt Godels incompleteness theory.



Paul Davies: Mathematics is, I think, for most people, the most reliable, the surest path to truth. We squabble over things like history and economics and politics and even medicine, but nobody would squabble over statements like '11 is a prime number', for example, it's simply true. So this idea that mathematics is a vast and elaborate network of statements that are either true or false is the one that most people carry around. Now what Gödel showed is that this tidy division into true and false based on the foundations of logic was a dichotomy built on sand.
Pauline Newman: It was in 1931 that Gödel dropped a bombshell in the foundations of mathematics and logical reasoning when he published his notorious incompleteness theorem, one of the weirdest mathematical papers ever. Its roots go back to ancient Greece. Paul Davies.
Paul Davies: Mathematics is built up step by step in the mathematical sequence. You start out with statements which seem to be self-evidently true which everybody accepts, and these are called axioms. I'll give you a simple example from geometry; between any two points in space it must be possible to draw a straight line. From a collection of statements like that you can build up ever more elaborate theorems. One that we learn at school is Pythagoras's theorem, and we accept that Pythagoras's theorem was true because we accept the axioms on which it's based, and every step in between is a logical step, it's a provable step, and that's why we can be so sure of mathematical theorems.
Pauline Newman: So far so good, but the Greek philosophers, steeped in the tradition of logic and rational reasoning, also caught a glimpse of a disturbing problem that over 2,000 years later was to completely demolish this tidy view of reality. The trouble concerned the existence of logical paradoxes.
Paul Davies: Suppose I say 'this statement is a lie' and then ask the question, 'Is the statement true or is it false?' Well, if the statement is true then it's a lie, so it's false. But if it's false, then it's true, and we get self-contradictory nonsense. For centuries this was just an amusing game among philosophers but what Gödel showed was that it infects mathematics as well, undermining its apparently solid logical foundations.
Pauline Newman: So how do they do that?
Paul Davies: Let's consider a simple statement like 'in the first trillion digits there are exactly six million prime numbers'. Is that true or is it false? Well, there's a simple mechanistic, handle-turning way of finding out the answer. I bet it's false but I don't happen to know. Mathematicians used to assume that all mathematical statements were like this; they were either true or false. Not so. What Gödel showed is that there exists mathematical statements that are simply undecidable. That is, that they are statements that might be true but they simply cannot be proved to be true, and there's no general procedure for knowing in advance which statements are undecidable and which are true and which are false.
Before Gödel's work, mathematicians assumed that truth and provability went hand in hand. That is, if a mathematical statement was true then in principle you could prove it, and if you could prove a particular theorem then it must be true. But after Gödel, this tidy division of mathematics into a vast and elaborate collection of true mathematical statements was swept away. Mathematics is now revealed not to be a closed system of truth but an open-ended logical mess.

This must be cheating a little. He proved the limitations of logic, but didn't actually discover something itself in this theory, like relativity, gravity, etc.

[Jorghnassen]

Bayes' rule!

/not a Bayesian myself...

[Unnamed]

According to this (http://www.abc.net.au/rn/scienceshow/stories/2006/1807626.htm#transcript), Kurt Godels incompleteness theory.
I bet he can't prove it.

[DRBUZZ0]

Greatest discovery in history? Um... that depends on how narrow your definition of discovery is. For example, would it be limited to a single item? Or to a whole field?


I would have to say the greatest single discovery was Newtonian Physics (laws of motion, gravity, kinetics ect). I suppose calculus would go along with that, as Newton really laid the foundation for both.

Some of the principals owe their roots to earlier mathematicians, but the creation of the whole unified system pretty much opened humanities eyes to how the world operates and set the stage for almost all mechanics, physics and technology that would come after.

Even if we've since discovered it has it's limitations (on quantum and astronomical levels)

[rjh01]

If you want the greatest discovery, how about the scientific process?

An alternative is the combination of paper and the printing press and other related discoveries. This allowed books to be published cheaply. This lead to many people learning to read, thus getting some sort of education.

Also it had great religious significance.

[SezMe]

I think the "scientific process" is an excellent candidate. Maybe "place value" numerical systems that permit the effective quantification of everything else.

[skeptifem]

i would say discovering bacteria and viruses... just wow. changed everything.

[Ginarley]

Writing.

[wipeout]

Richard Feynman talked about the huge importance of the knowledge roughly summed up as "all things are made of atoms".

I'd go with that.

[Admiral]

There's some serious competition on this thread, but I'm going to go with calculus.

It's impossible to get far in almost any area of physics without using calculus, and it finds itself deeply entangled in many other sciences as well.

[Mid]

I'd be tempted to go for fire, although is that going back a bit to far?

[a_unique_person]

Don't forget, Gödel has umlauts.

[Soapy Sam]

Beer.

[Dave1001]

There's some serious competition on this thread, but I'm going to go with calculus.

It's impossible to get far in almost any area of physics without using calculus, and it finds itself deeply entangled in many other sciences as well.

well then, along those lines, how about arithmatic?:p

[TragicMonkey]

Agriculture. Without agriculture, humans would never have been able to stay in the same place long enough to develop the leisure to discover anything else.

[l0rca]

I second beer.

[CFLarsen]

The scientific method. Sans comparison.

[Soapy Sam]

well then, along those lines, how about arithmatic?:p
Hwo abuot speling?

[CFLarsen]

Hwo abuot speling?

I've seen an in-between-programme blurb on Discovery, where a text was typed on screen, where just about every word was deliberately misspelled. Only the first and last letter of each word were in the right place, but it was still clear what the message was.

Namely, that it was possible to understand a text, even if it was spelled that way.

Cool! :)

[Dancing David]

String and glue.

[RenaissanceBiker]

Beer.

Well, the day I discovered beer was pretty momentous. However, for society as a whole, I'm going to have to go with Calculus. I think the heliocentric model of the solar system deserves an honorable mention.

[aggle-rithm]

Cooking.

Cooked meat allowed for higher protein consumption, which created the environment necessary for larger brains to evolve.

[H3LL]

No contest...

It's got to be zero.



Or was that invented?

.

[Marc L]

Hwo abuot speling?

Actually, I'd say that'd be covered under typing, as none of the words (except the last) are mis-spelled, just mis-typed.

As for my two cents, I'd say, without a doubt, that the microwave was the greatest discovery in history.

Marc

[Mosquito]

I'll have to go with photosynthesis. It allowed enough Oxygen to stay in the atmosphere for the evolution of animals. And animals need a system for distribution of this Oxygen to all the little cells in their bodies. And blood beats beer anytime.

Mosquito - whaddayamean "human" history? Bloody(yum) self-centered newcomers

[RenaissanceBiker]

Ok, here's my top 10 greatest discoveries in human history. Any non-humans that want to complain about being left out can do so by PM.

10. The Wheel/Tools. The wheel's impact is indisputable since we still rely on it heavily today. Making stone tools is now only a hobby for native-american enthusiasts, but it had a tremendous impact on our ancestor's ability to thrive in a hostile world.

9. Fire. For warmth and cooking it is without equal. I still heat my home and cook most of my food with it.

8. Agriculture/Beer. Having a large food plot outside your door, rather than foraging far and wide for sustenance, turned out to be a great time and energy conserving strategy. It worked for dometicating animals too. Then someone stumbled across beer. That really brought people together and encouraged them to work together in order to raise enough grain to produce it.

7. The Lever. The ability to lift and move heavy things helped build massive structures. it also helped knock them down on occasion.

6. Written Language. This is the ability to communicate across time, although in only one direction. It can also be used to communicate the same idea to more than one person, at different times. That's a big improvement over oral communication.

5. Arabic Numbers. Roman numerals were ok for record keeping, but they sucked for performing calculations. Arabic numbers were a big improvement. Personally, I wish they had started with a base-12 number system but I'll work with what we have.

4. Algebra/Geometry. This took mathematics beyond simple operations. It allowed us to make general calculations that had many applications. It became a system for making new mathematic discoveries, that expanded the field of mathematics, that lead to new discoveries, etc.

3. Cell Theory of Biology/Evolution. This is the foundation of modern medicine, the basis of understanding who and what we are and how to live better and longer.

2. Thermodynamics. Before this, we knew of fire but we didn't know about fire. After this, we learned how to harness the basic power of the universe.

1. Calculus/Newtonian Physics. This combination sparked the industrial revolution in the western world. It provided a sound basis for scientists and engineers to examine and make predictions about the world around them.

Dang, I still didn't put the Heliocentic Model of the Solar System in there! I'd like to work Hydraulics, Chemstry/Atomic Model, Relativity, Electo-Magnetic Theory and Semi-conducters in there as well.

[Big Al]

For me, writing stands head, shoulder, waist and knees above any other invention or discovery. Until writing came along, people's knowledge was limited to what they had personally experienced, or by what they were directly told by someone else alive at the time. Ever heard of Chinese Whispers?

Writing allows us to learn directly from people all over the world or long dead. Early science didn't stand a chance without access to prior records of experiments, observations and results. Without writing, how would calculus, Gödel's Incompleteness Theorem, Schrödinger's Equations or any other scientific truth ever have come about, let alone have been passed on to future generations?

Writing every time for me. It unlocked human progress unlike anything else..

[pgwenthold]

the semiconductor

[davefoc]

If we're looking for the discovery that had the greatest impact on the civilization has we know it, organized warfare might be the original driving force.

Most technology was originally driven by it, without organized warfare we wouldn't be human as we know the concept today.

[Dave1001]

I've seen an in-between-programme blurb on Discovery, where a text was typed on screen, where just about every word was deliberately misspelled. Only the first and last letter of each word were in the right place, but it was still clear what the message was.

Namely, that it was possible to understand a text, even if it was spelled that way.

Cool! :)

I saw that too (on the internet) but then I read somewhere that the example deliberately designed to work -that in most cases something like that wouldn't be so readable. It would be interesting to find out the truth about that.

[Kaylee]

I agree that writing is the most important discovery. It's essential for preserving knowledge . Combined with wide spread literacy and easier and less expensive ways of printing (leapfrogging from handwritten manuscripts to the Gutenberg printer, electronic printers, faxes and the Internet) it also can prevent knowledge from being monopolized by the privileged few.

We recently had several threads in this forum about a 2,000 year old mechanical computer, the Antikythera Mechanism (http://www.cbc.ca/technology/story/2006/11/30/antikythera-mechanism.html), whose purpose and inner workings were only recently understood even though it had been recovered from a shipwreck over a 100 years ago. Scientists were surprised that this degree of technology had existed so long ago. What are the chances that the knowledge of how to build a specialized piece of scientific equipment would be lost today? More things are written down and more people are literate and have the knowledge that comes with literacy.

I agree with aggie-rithmn and Tragic Monkey that cooking and agricultural were also important discoveries, and without those discoveries its unlikely that we would have gone on to ever discover anything else that was important.

But I think its possible that the discoveries of cooking and agriculture (especially agriculture) could have been lost as a result of natural disasters. I vote for writing because of its side-effect of preserving knowledge.

[Ian Osborne]

No contest...

It's got to be zero.

Dang, you beat me to it. Calculus is itself dependent on an awareness of 'zero', and it's such a fundemental part of pure mathematics, I'd say it was definitely discovered rather than invented.

[drkitten]

Dang, you beat me to it. Calculus is itself dependent on an awareness of 'zero',

I'm not sure it does. Remember Archimedes and his "method of exhaustion"?

And, of course Archimedes had no notion of zero.

[robinson]

I swa taht too (on the intrenet) but tehn I raed somwhre that the exmaple wsa delbieratly desegnid to wrok -that in msot csaes somithnig lkie that wuoldn't be so raedable. It wuold be inetersting to fnid out the trtuh abuot taht.

heh

[baron]

Quantum entanglement will, I believe, form part of the greatest discovery ever.

I also rejoice in the fact nobody will agree with me :D

[Jimbo07]

I agree with aggie-rithmn and Tragic Monkey that cooking and agricultural were also important discoveries,

I vote for writing because of its side-effect of preserving knowledge.

Without a building to write in, you'd be reduced to painting prehistoric graffiti on cave walls...

Regression:
Some of these aren't one discovery (in the modern sense that we can credit individuals). It may be that several groups harnessed lightning fires about the same time... I don't know, and I'm not sure how anyone would find out (campsites with the earliest fire pits?).

Who was the first to till soil? It may be traceable to a group, but an individual?

...

Similarly, my vote like on the science thread, is for a history of assembled discoveries, namely: engineering. From the earliest civil structures and fortifications, through Roman plumbing, the mechanical engineering of great seige weapons, the secrets of Damascus steel and through to the modern ideals of science-based electrical and chemical engineering, I vote for engineering.

Whether the tradition was preserved by Roman Army engineers, Stone Masons, or modern university trained engineers, I'll recap:

Without a building to write in, you'd be reduced to painting prehistoric graffiti on cave walls...

;)

(Heh... this reminds me of those 'person of the millenium' lists... can't we all just agree to disagree? :D )

[RenaissanceBiker]

I couldn't even agree on my own top ten!

[H3LL]

I'll take back my zero I have decided the greatest discovery must be finding out what you can do with female genitalia.

What's more new people are making this greatest of discoveries every day.

[RenaissanceBiker]

That and beer are personal discoveries.

[H3LL]

That and beer are personal discoveries.

You're going to the wrong parties.


:D

[Iamme]

Plain ol electrical current; ac and dc

[Ben Tilly]

5. Arabic Numbers. Roman numerals were ok for record keeping, but they sucked for performing calculations. Arabic numbers were a big improvement. Personally, I wish they had started with a base-12 number system but I'll work with what we have.

Useless trivia.

The reason why Roman numerals sucked for performing calculations is that nobody thought of them as a way of doing calculations. Instead people did calculations on an abacus then wrote down the answer. In fact historians have documented that the spread of Arabic notation is directly tied to the decline of the abacus.

Anyways the invention that I consider very important is the idea of a phoenetic alphabet. Invented, of course, by the Phoenecians. It was particularly useful for them because they traded with people using different languages and odd names. The Phoenecians could write down notes that explained how to pronounce names and what useful phrases were for trade, nobody else could.

Cheers,
Ben

[Schneibster]

I gotta go with language. Spoken language has led to everything else we've ever done, including considering what the greatest discovery in history was.

"Beer" did, however, win at least an emotional vote from me before I seriously considered the question.

[Starthinker]

I think the coolest discovery was that when I drove my Oldsmobile in the winter the air turbulence around the hood ornament would form, under the perfect conditions, a little ball of snow that would "hover" right behind the hood ornament. Kept me amused for hundreds of miles.

[Jeff Corey]

The wedge. A simple tool. Evolved into inclined planes, pointed sticks,knives,screws, a variety of useful implements. The rock was good to have around, too. And the club. Or the rock tied to the end of the club.
And thermonuclear devices.

[wipeout]

Quantum entanglement will, I believe, form part of the greatest discovery ever.

I also rejoice in the fact nobody will agree with me :D

Only part? Not sure what you mean. Maybe something to do with teleporters? :D

Entanglement is the reason individual objects exist and keep existing without quantum interference so it's a somewhat important discovery on its own anyway as, without that, everything goes seriously Schrodinger's cat. :)

[rjh01]

I think that there is no one 'greatest discovery in history' Any one discovery will advance humans a certain amount. Then you need something else to advance humans more. Take fire as an example. It meant you could live in cooler climates. It meant that we could cook meat and veg. The population numbers went up. However we were still hunter gathers. Agriculture meant that humans could have more possessions than what could be carried (plus see above). Population numbers went up. Cities were created, people became specialists.

Each discovery depended on the previous ones. Until they were made little changed. The Romans discovered as much as they could short of the industrial revolution. They lacked cheap, mass produced books to educate the masses. It was these people that created the industrial revolution.

One discovery has revolutionised the 1950 - present and no-one has mentioned it. I refer to the computer. Without it we would still be employing many low paid workers to do payroll. Government services would only be very basic, as many taxes depend on computers. Living standards would be very much lower. Not to mention WW2 lasting another year or so (or worse).

[The Atheist]

H3LL's nearly got it.

The greatest human discovery was that sex is good, and not just for procreation.

[Big Al]

I saw that too (on the internet) but then I read somewhere that the example deliberately designed to work -that in most cases something like that wouldn't be so readable. It would be interesting to find out the truth about that.

I raed in New Senicsitt reneclty that resaerch indiacets taht as lnog as the beingning and end lteters are perversed, you can mkae sesne of petrtry mcuh any senetcne whituot too mcuh torulbe

[SezMe]

I raed in New Senicsitt reneclty that resaerch indiacets taht as lnog as the beingning and end lteters are perversed, you can mkae sesne of petrtry mcuh any senetcne whituot too mcuh torulbe
Huh? Couldn't get that. :)

[Alphaba]

On a usefulness-desirability continuum I would choose: calculus - microbiology - agriculture - writing - music - Champagne wine.



However, as the highest on both dimensions, the all time winner unquestionably is... The Pill.

[Angus McPresley]

What Gödel showed is that there exists mathematical statements that are simply undecidable. That is, that they are statements that might be true but they simply cannot be proved to be true, and there's no general procedure for knowing in advance which statements are undecidable and which are true and which are false.


This just doesn't sound accurate to me. What Gödel showed is that, with any set of operators, that there were statements that could not be decided true or false with that set of operators. Not that they're "undecidable", but that their truth or falsehood can't be determined with a fixed set of operators. And what does it mean to know "in advance" which statements are undecidable?

Is it just me, or is he confused here? What's an example of an undecidable statement?

[Larry Lovage]

I'm not sure you're right, Angus. Having first demonstrated that any particular set of axioms and rules would generate an undecidable theorem. Any modification of the axioms and rules to exclude this undecidable theorem, would in fact be a new set of axioms and rules, which would in its turn generate its own undecidable theorem.

The first undecidable theorem that forms part of Gödel's proof is: "This statement has no proof." It's true, as any of us can see. But it can't be proved.

Any unproved mathematical hypothesis might be undecidable. We don't know. It's possible that Goldbach's Conjecture, an unproven mathematical statement, is undecidable. Also the Riemann Hypothesis.

The trouble is that the OP quotation has these two people seemingly convinced that after Gödel, mathematics was meaningless. All it actually meant was that Hilbert's hope for a complete system of axioms and rules, (one such was being worked on at the time by Russell and North Whitehead, known as Principia Mathematica or PM for short) was no longer possible. This was a crashing blow for the mathematical hopes of completeness in the early 20th Century, but it certainly didn't wipe out all the proofs which had been found (without those proofs, incidentally, Gödel's theorem means absolutely nothing).

[H3LL]

The greatest human discovery was that sex is good, and not just for procreation.

Pros evolved, they weren't created. :mad:



I'm off to the warehouse....






;)
.

[boooeee]

H3LL's nearly got it.

The greatest human discovery was that sex is good, and not just for procreation.
Didn't it happen the other way around? You think the cavemen and cavewomen were bumping uglies for the good of the species? :wink:

Back to the OP, it seems that the importance of Godel's theorem is being greatly exaggerated. I admire it as one of the most original pieces of mathematical/logical thinking in history. However, it seems more of a curiosity than anything else.

The only application of Godel's theorem I know of is that somebody proved that some theorem about infinity (one of the Cantor varieties) was "formally undecidable" in the Godel sense.

Maybe I'm just ignorant on this, but has Godel's theorem had an impact on people, outside of academic logicians and popular science writers?





And duh, it's beer. With quesadillas (the world's most perfect food) coming in at a close second.

[asthmatic camel]

Duct tape.

[Jorghnassen]

H3LL's nearly got it.

The greatest human discovery was that sex is good, and not just for procreation.

That might've been true if only the human species had "discovered" this. Unfortunately, bonobos (and dolphins, I think), also do the sex for pleasure without procreative purpose thing.

/the greatest discovery in the world, in fact, is the concept of run-on sentences that are also vague and imprecise

[Ben Tilly]

This just doesn't sound accurate to me. What Gödel showed is that, with any set of operators, that there were statements that could not be decided true or false with that set of operators. Not that they're "undecidable", but that their truth or falsehood can't be determined with a fixed set of operators. And what does it mean to know "in advance" which statements are undecidable?

Is it just me, or is he confused here? What's an example of an undecidable statement?

That isn't quite what Gödel showed.

He showed that, within any finite axiom system that can model the positive integers, if it proves that it is consistent, then it is inconsistent. (ie From a proof of consistency you can generate internal contradictions.) Therefore if the axiom system is going to be consistent, then the consistency of the axiom system must be undecidable within that axiom system.

Therefore, for instance, within the Peano axioms one would hope the statement that the Peano axioms are consistent is undecidable. We all believe the consistency of the Peano axioms, but the only way to prove it is to make an assumption that implies it.

Now this is merely the first instance of an undecidable statement. Many others exist. For instance there is a polynomial in 42 variables which the Peano axioms (hopefully) cannot prove has any integral solutions.

Now the undecidable statements that we can find are all very artificial. But their existence raises the question of whether there are interesting mathematical conjectures that are truly undecidable. There may be, but obviously we cannot identify which ones they are.

That was a starting point. People have refined this result in various ways. Chaitin in particular has, see http://www.cs.auckland.ac.nz/CDMTCS/chaitin/ for a list of papers. Let me give just one example from http://www.cs.auckland.ac.nz/CDMTCS/chaitin/lisp.html. Let's say an elegant program is a program such that no shorter program always produces the same output. Well it is easy to show that, for any language, there are an infinite number of elegant programs. What is not so easy to show is that, for any language and axiom system, only a finite number of elegant programs can ever be proven to be elegant.

That isn't exactly what he does in that paper. Instead he describes a simple form of Lisp, then shows that for any consistent axiom system, no elegant program written in that version of Lisp which is more than 410 characters longer than a description of the axiom system can be proven to be elegant. He indicates that (and it is fairly straightforward to show) a similar result is true in any language, but the constant 410 will vary. (Normally it will be much bigger, he chose his version of Lisp to make it easy to make it small.) And since only a finite number of elegant programs are below that size limit, only a finite number of the infinite number of elegant programs can be shown to be elegant.

Anyways there are lots of undecidable problems. We can show that. We can find problems that we can show are undecidable. There are bound to be interesting problems that are undecidable. But we don't know which ones they may be.

Cheers,
Ben

[Schneibster]

To put it another way, what Gödel showed was that no axiomatically-based system can be both complete- that is, capable of proving the truth or falsehood of all statements that can be formed in that system- and consistent- that is, incapable of proving the truth of two statements that are mutually contradictory. Most mathematicians prefer to use the incomplete version of number theory rather than the inconsistent one, which makes a certain amount of sense if you think like a mathematician.

The more interesting corollaries of this are, first, the type of logic that we use is based on the same type of reasoning and subject therefore to the same flaws, i.e., "this statement is a lie" is undecidable, and there is a two-step inconsistent pair of statements based on the same idea, and second, we describe physics using mathematics and no one I am aware of has seriously considered the implications of Gödel's Theorem on that math.

[The Atheist]

That might've been true if only the human species had "discovered" this. Unfortunately, bonobos (and dolphins, I think), also do the sex for pleasure without procreative purpose thing.Dolphins?

You're telling me that with all those teeth dolphins practice fellatio?

[a_unique_person]

The first undecidable theorem that forms part of Gödel's proof is: "This statement has no proof." It's true, as any of us can see. But it can't be proved.


Maybe that's why he went nuts. He was convinced people were trying to poison him. There is no way you can prove absolutely that no-one was trying to poison him. Although he did end up starving himself to death because he wouldn't eat. Is that some sort of logical fallacy he committed on himself?

[Iamme]

I think the coolest discovery was that when I drove my Oldsmobile in the winter the air turbulence around the hood ornament would form, under the perfect conditions, a little ball of snow that would "hover" right behind the hood ornament. Kept me amused for hundreds of miles.

Water beads climbing UP my windshield has always mesmerized me, also, while driving down the road.

[Iamme]

The wedge. A simple tool. Evolved into inclined planes, pointed sticks,knives,screws, a variety of useful implements. The rock was good to have around, too. And the club. Or the rock tied to the end of the club.
And thermonuclear devices.

And wood chippers. Just think...if some poor country can't afford to pay someone for nuclear weapons technology or the ingredients, they could always get a bunch of wood chippers and throw enemies into those...simply for a change of pace, from their usual lopping of people's heads off.

Edit: I just read this to ecman and he thought it be an excellent device to use for the death penalty. :)

[Gord_in_Toronto]

The horse/buffalo/ox collar. Made larger scale farming possible. Made it possible one farmer to feed more than just his own family, led to the differention of trades, free time and made civilization (such as we know it) possible.

[SquishyDave]

A group of my work mates tested the mixing up the middle of words thing, and most things quickly became illegible if the middle letters where truly randomised. One of them threw together a quick program to test it. Try it yourself.

Also, no one has mentioned the flush toilet have they? Damned fine invention.

[Ian Osborne]

The horse/buffalo/ox collar. Made larger scale farming possible. Made it possible one farmer to feed more than just his own family, led to the differention of trades, free time and made civilization (such as we know it) possible.

Good one

[rjh01]

Also, no one has mentioned the flush toilet have they? Damned fine invention.

You are not too far off. When you get a large city and poor hygiene you get serious decease. So you need a decent sewerage system of which flush toilets are just a small part.

Study the London sewerage system (http://en.wikipedia.org/wiki/London_sewerage_system) for more information on what I mean.

[3point14]

Greatest discovery ever?

Perhaps trade. Without the ability to trade, a vast number of people who made significant discoveries since would have been too busy finding/raising/rearing food and building shelters to do much discovering.

[rjh01]

Other great inventions
Mass production. This made goods a lot cheaper.

Motor car. Changed cities a lot. Where to go for a holiday. Put horses out of a job.

Certain drugs such as vaccines and penicillin. Reduced sick leave for children so they could go to school more often.

Flying. Now the middle class could afford to go to the other side of the world and do things when they get there. Made regular meetings of world leaders possible.

Internet. Made this forum possible. So much easier to find facts. Almost makes reference books obsolete.

Explosives. Made modern warfare possible. Made mining a lot cheaper.

[drkitten]

I'm not sure you're right, Angus. Having first demonstrated that any particular set of axioms and rules would generate an undecidable theorem. Any modification of the axioms and rules to exclude this undecidable theorem, would in fact be a new set of axioms and rules, which would in its turn generate its own undecidable theorem.

The first undecidable theorem that forms part of Gödel's proof is: "This statement has no proof." It's true, as any of us can see. But it can't be proved.

But that's Angus's point. The statement is easily provable. Were it false, then the statement would, in fact, have a proof. Assuming the consistency of the system under discussion, then that would make a false statement provable, violating the consistentcy assumption. Therefore the statement is true.

Proof by contradiction. I just used a different system to do it.

Godel showed that for any (consistent) system, there exist statements that that system cannot prove. That's not the same thing as showing that the statement can never be proved. I can always design another system in which the statement is an axiomatic truth -- or axiomatic falsehood, for that matter.


Any unproved mathematical hypothesis might be undecidable. We don't know. It's possible that Goldbach's Conjecture, an unproven mathematical statement, is undecidable.

No, it's not. I'm currently working in a six-axiom system, which I call the Peano-Drkitten system. The first five axioms are the Peano axioms, the sixth is the truth of the Goldbach conjecture. I've demonstrated some interesting properties of this system already (did you know that addition is commutative in the Peano-Drkitten system?), but I haven't yet been able to prove that the system is consistent....

[Oxymoron]

I don't know if this is a discovery, but what about critical thinking? Without it, we probably wouldn't have many of the discoveries listed on this thread. Just a thought.:)

[69dodge]

This just doesn't sound accurate to me. What Gödel showed is that, with any set of operators, that there were statements that could not be decided true or false with that set of operators. Not that they're "undecidable", but that their truth or falsehood can't be determined with a fixed set of operators.I'm not sure what you mean by "operator" here. Can you explain?

[69dodge]

It's possible that Goldbach's Conjecture, an unproven mathematical statement, is undecidable.

No, it's not. I'm currently working in a six-axiom system, which I call the Peano-Drkitten system. The first five axioms are the Peano axioms, the sixth is the truth of the Goldbach conjecture. I've demonstrated some interesting properties of this system already (did you know that addition is commutative in the Peano-Drkitten system?), but I haven't yet been able to prove that the system is consistent....I don't get your point.

You say, "No, it's not." Do you think you're actually contradicting the statement you quoted?

Surely, its author didn't mean, "it's possible that Goldbach's Conjecture is undecidable in the Peano-Drkitten system", because, first of all, he never heard of that system because you just made it up, and second of all, why should anyone care about what can be "proven" in a system that we have no reason to believe is consistent, especially when the proof consists of saying, "it's an axiom". I can "prove" anything if I'm allowed to do it that way.

[drkitten]

I don't get your point.

You say, "No, it's not." Do you think you're actually contradicting the statement you quoted?

Surely, its author didn't mean, "it's possible that Goldbach's Conjecture is undecidable in the Peano-Drkitten system",

The author didn't mean what he wrote, because he had no idea what he wrote. And there's no circumstance under which a well-informed person would have written the sentence that he did, because there's no interpretation under which Godel's theorem is at all signficant.

Let's look at the Goldbach conjecture in detail. Lovage wrote, "It's possible that Goldbach's Conjecture, an unproven mathematical statement, is undecidable."

There are two possible interpretations of this statement, both of which are trivial cases.

Case 1 : he meant that "there exists a system under which Goldbach'c conjecture is undecideable." This is trivially true; any system with no rules of inference that doesn't include Goldbach's conjecture as an axiom would do it.

Case 2 : he meant that "there does not exist a system under which Goldbach's conjecture is decideable." This is trivially false -- any inconsistent system, or system in which Goldbach's conjecture is an axiom would do it.

Lovage -- and you -- clearly do not understand the significance of Godel's theorem(s). They're not about statement being decidable or undecideable, but about systems being capable of making decisions. There is no such thing as an "undecidable" statement except in the context of a particular system for deciding.


I can "prove" anything if I'm allowed to do it that way.

... which is why Kurt Godel didn't try to prove anything like "there exist unprovable statements." Godel was well aware that any system can be patched to allow anything to be provable as an axiom (in fact, that's the second half of his First Incompleteness Theorem paper, showing that this process still does not allow a "perfect" proof system to be generated).

[Tez]

T
Now the undecidable statements that we can find are all very artificial. But their existence raises the question of whether there are interesting mathematical conjectures that are truly undecidable.


Homeomorphism of topological spaces is undecidable, and is the most fundamental question in topology - so I'd think that counts that as interesting. Well, to those pretty boring topologists anyway :)

[Ben Tilly]

The author didn't mean what he wrote, because he had no idea what he wrote. And there's no circumstance under which a well-informed person would have written the sentence that he did, because there's no interpretation under which Godel's theorem is at all signficant.

Let's look at the Goldbach conjecture in detail. Lovage wrote, "It's possible that Goldbach's Conjecture, an unproven mathematical statement, is undecidable."

I hate it when people have more technical knowledge than common sense. And then say overly nasty things that aren't true.

I consider myself a well-informed person on this topic. And I could easily see myself saying, "It's possible that Goldbach's Conjecture, an unproven mathematical statement, is undecidable." What I would mean is "undecidable in any widely accepted set of axioms". If I was pushed farther I'd say ZFC. (Though C is irrelevant here, accepting or not accepting choice cannot change any results in number theory.)

There are two possible interpretations of this statement, both of which are trivial cases.

Case 1 : he meant that "there exists a system under which Goldbach'c conjecture is undecideable." This is trivially true; any system with no rules of inference that doesn't include Goldbach's conjecture as an axiom would do it.

Any self-respecting mathematician would insist on only talking about Goldbach's conjecture from within an axiom system that is sufficiently rich to model the positive integers. Else the phrase "Goldbach's conjecture" is just a meaningless phrase without the semantic content that mathematicians associate with that phrase. That makes case 1 very non-trivial. It may be that there is a proof we have not found of Goldbach's conjecture from the Peano axioms. In which case there is no system under which Goldbach's conjecture is undecidable.

Case 2 : he meant that "there does not exist a system under which Goldbach's conjecture is decideable." This is trivially false -- any inconsistent system, or system in which Goldbach's conjecture is an axiom would do it.

I note that you've missed the common sense case. Which is that it is undecidable from within the axioms that mathematicians commonly use.

Lovage -- and you -- clearly do not understand the significance of Godel's theorem(s). They're not about statement being decidable or undecideable, but about systems being capable of making decisions. There is no such thing as an "undecidable" statement except in the context of a particular system for deciding.

It is possible to possess an exact understanding of the theorem yet say things in a slightly sloppy way. Not only is it possible, it is standard practice.

... which is why Kurt Godel didn't try to prove anything like "there exist unprovable statements." Godel was well aware that any system can be patched to allow anything to be provable as an axiom (in fact, that's the second half of his First Incompleteness Theorem paper, showing that this process still does not allow a "perfect" proof system to be generated).

When you're writing down theorems and coming up with proofs, you need to be precise. When you're discussing ideas, people are seldom precise. Live with it.

Cheers,
Ben

[Ben Tilly]

Homeomorphism of topological spaces is undecidable, and is the most fundamental question in topology - so I'd think that counts that as interesting. Well, to those pretty boring topologists anyway :)

Um, let's go back one level of meta.

It is true that the problem of proving that topological spaces is undecidable. But that does not mean that deciding whether a given pair of topological spaces is homeomorphic is necessarily undecidable.

Figuring out what programs will do when run (*) is likewise undecidable. But that does not mean that deciding what a given program will do when run is necessarily undecidable.

My comment was that we don't know if there are any interesting mathematical conjectures that are undecidable. By that I meant specific things that we'd like to know but can't. For instance there might be a pair of topological spaces that we'd want to know is homeomorphic. Or we want to know what a specific program will do when run. Or we'd like to know if Goldbach's conjecture is true. Or whatever.

Cheers,
Ben

* This is only true if the program is written in a Turing-complete language. Most, but not all, languages are Turing-complete. Useless trivia time, the most widely used language that is not Turing-complete is SQL. It is deliberately not Turing-complete to make it easier for databases to optimize.

[69dodge]

Let's look at the Goldbach conjecture in detail. Lovage wrote, "It's possible that Goldbach's Conjecture, an unproven mathematical statement, is undecidable."

There are two possible interpretations of this statement, both of which are trivial cases.

Case 1 : he meant that "there exists a system under which Goldbach'c conjecture is undecideable." This is trivially true; any system with no rules of inference that doesn't include Goldbach's conjecture as an axiom would do it.

Case 2 : he meant that "there does not exist a system under which Goldbach's conjecture is decideable." This is trivially false -- any inconsistent system, or system in which Goldbach's conjecture is an axiom would do it.I can think of another, much more interesting, interpretation: it's possible that Goldbach's conjecture isn't decideable in the more or less vaguely defined "system" that consists of the methods of proof which are generally recognized as such by today's mathematicians.

[ETA: as independently discovered by Ben Tilly while I was typing :D]

Lovage -- and you -- clearly do not understand the significance of Godel's theorem(s).I won't speak for him, but I think I understand it pretty well.

Clearly, it isn't clear that I do, as you got the impression that I don't; however, if you think it's clear that I don't, your definition of clarity lacks the desirable property that clarity implies truth. :p

They're not about statement being decidable or undecideable, but about systems being capable of making decisions. There is no such thing as an "undecidable" statement except in the context of a particular system for deciding.Ok, so I guess this was your point, then.

I agree with it.

(Surprise!)

[Iamme]

The whole shipping-trucking-grocery stores business. If it weren't for this, we would either have to live by farms or live ON one.

[Ben Tilly]

[ETA: as independently discovered by Ben Tilly while I was typing :D]

I suspect that the order was the other way around. It was independently discovered by you while I was typing.:p

Cheers,
Ben

[Larry Lovage]

Excellent work defending me, Ben. However, I had read several well informed posts prior to drkitten getting on my case and had recognised instantly better representations of Gödel's theorem than I had provided. I think I agree with drkitten that what Gödel was proving was about axiomatic systems, not about provability.

What I don't understand about drkitten's statements is the following:No, it's not. I'm currently working in a six-axiom system, which I call the Peano-Drkitten system. The first five axioms are the Peano axioms, the sixth is the truth of the Goldbach conjecture. I've demonstrated some interesting properties of this system already (did you know that addition is commutative in the Peano-Drkitten system?), but I haven't yet been able to prove that the system is consistent.... I get that this is a joke. But isn't it kind of against the rules of mathematics to simply add a new axiom to cover unprovable stuff? Surely there is a real world of mathematics based on the minimum axioms that we can see to be true. I don't think Gödel was trying to specifically rule out further axiomatic systems, only that they wouldn't help. But again, that's not what I meant by "being against the rules".

I'm happy to learn, particularly if I've been egregiously wrong. (Just, er, don't call me "Lovage", ok?)

[drkitten]

Excellent work defending me, Ben. However, I had read several well informed posts prior to drkitten getting on my case and had recognised instantly better representations of Gödel's theorem than I had provided. I think I agree with drkitten that what Gödel was proving was about axiomatic systems, not about provability.

What I don't understand about drkitten's statements is the following:I get that this is a joke. But isn't it kind of against the rules of mathematics to simply add a new axiom to cover unprovable stuff?

Not at all. In fact, "adding a new axiom to cover unprovable stuff" was one of the questions specifically addressed in Godel's original work.

More to the point, there are at least two well-understood historical examples of this : ZFC and the GCH. The system of formal set theory espoused by Whitehead and Russel in Principia is almost impossible to work with, and most modern mathmaticians prefer to work in a simpler and cleaner system called ZF (after its inventors, Zermelo and Fraenkel). However, it was discovered that a lot of interesting proofs required not only ZF but an additional, unprovable assumption called the "Axiom of Choice."

There was a tremendous debate in the literature about whether or not the Axiom of Choice was acceptable, especially after it was proven to be "independent" of (meaning unprovable by) The rest of ZF. So there are actually two competing systems of set theory out, ZFC and "pure" ZF, although today ZF has more or less lost the popularity contest.

GCH -- the "Generalized Continuum Hypothesis" -- has a similar story. We can prove that we can neither prove it true or false using more "standard" mathematics (in particular, the GCH is independent of ZFC). So do we assume it or not?


Surely there is a real world of mathematics based on the minimum axioms that we can see to be true.


But that's the question. How about things that make sense, but that we can't necessarily "see" to be true?

For example, the GCH implies that there are no infinite cardinals between the size of the set of integers and the size of the set of real numbers. I find that plausible and elegant -- but is it "true"? What would you look at to "see" whether or not that's true? The axiom of choice states that any set -- even an infinite set -- can be well-ordered. Again, I find this plausible, but is is "true"?

[Mangafranga]

A few comments Re Godel.
As I understand it his proof applies only to systems with a finite set of axioms.
The reason inconsistency is a major problem is that in classical logic from a contradiction you can deduce any proposition you like. There are logicians that think that this aspect of classical logic is not correct, see paraconsistent logic and inconsistent mathematics.

[Ben Tilly]

Excellent work defending me, Ben. However, I had read several well informed posts prior to drkitten getting on my case and had recognised instantly better representations of Gödel's theorem than I had provided. I think I agree with drkitten that what Gödel was proving was about axiomatic systems, not about provability.

What Gödel proved was about provability within axiom systems. However virtually no real mathematicians ever think about the axiom system they are working in - they just use whatever is standard in their field. Most know enough to know that, somehow and somewhere, this goes back to set theory. Most in turn know that the important axioms are called ZFC. But most couldn't name any of the axioms of ZFC other than choice.

Therefore for most mathematicians "provability" really means "provability within a default axiom system".

But number theory gets more interesting. You see, we can prove things such as, "If ZFC is consistent then this particular polynomial in 42 variables which the Peano Axioms cannot prove nor disprove has integral solutions." And the logicians will tell us, "This means that there are models of the integers in which this has solutions, and ones in which it doesn't."

But wait a minute. Either a given set of integers is or is not a solution of this polynomial. How can there be uncertainty about that?

Well what this really means is that there are models of the Peano axioms that have things in them that somehow "really" aren't integers. Anything that we'd agree are "really" integers cannot be a solution of this polynomial. So, even though the Peano Axioms are silent on the existent of integral solutions, there "really" aren't solutions. (I've put really in quotes because it is a concept that cannot be formalized in first order logic, but it is a concept that we all can intuitively grasp.)

When mathematicians talk about integers, they are formally talking about things that fit axioms, but they are trying to talk about "real" integers.

What I don't understand about drkitten's statements is the following:I get that this is a joke. But isn't it kind of against the rules of mathematics to simply add a new axiom to cover unprovable stuff? Surely there is a real world of mathematics based on the minimum axioms that we can see to be true. I don't think Gödel was trying to specifically rule out further axiomatic systems, only that they wouldn't help. But again, that's not what I meant by "being against the rules".

I'm happy to learn, particularly if I've been egregiously wrong. (Just, er, don't call me "Lovage", ok?)

Mathematics doesn't really have rules. It is something we made up which seems to work pretty well. :D

The point of drkitten's joke is that you can propose anything as an axiom system. The consistency of that system is something you can never prove (except in the negative) - it has to be accepted or rejected on heuristic grounds. (Only true for axiom systems that model the integers, there are finite axiom systems that can be, and have been, proven to be consistent.) So yes, what she put down is an axiom system.

Cheers,
Ben

[Ben Tilly]

Not at all. In fact, "adding a new axiom to cover unprovable stuff" was one of the questions specifically addressed in Godel's original work.

More to the point, there are at least two well-understood historical examples of this : ZFC and the GCH. The system of formal set theory espoused by Whitehead and Russel in Principia is almost impossible to work with, and most modern mathmaticians prefer to work in a simpler and cleaner system called ZF (after its inventors, Zermelo and Fraenkel). However, it was discovered that a lot of interesting proofs required not only ZF but an additional, unprovable assumption called the "Axiom of Choice."

There was a tremendous debate in the literature about whether or not the Axiom of Choice was acceptable, especially after it was proven to be "independent" of (meaning unprovable by) The rest of ZF. So there are actually two competing systems of set theory out, ZFC and "pure" ZF, although today ZF has more or less lost the popularity contest.

For reasons that you'd understand if you tried to do real analysis without having Zorn's Lemma available. :-)

Mathematicians don't really care about Truth per se, they just care about being able to create pretty theories and results. Assuming choice allows them to get nicer theories and results, so everyone assumes choice.

GCH -- the "Generalized Continuum Hypothesis" -- has a similar story. We can prove that we can neither prove it true or false using more "standard" mathematics (in particular, the GCH is independent of ZFC). So do we assume it or not?

Who cares? Unless you're a logician, the continuum hypothesis isn't likely to make a huge difference.

But that's the question. How about things that make sense, but that we can't necessarily "see" to be true?

There is a split here. On the one hand there are things like number theory where we have an intuitive, if unformalizable, notion of what it means for something to be "really" true. On the other we have areas where mathematics is modelling something complex and arbitrary enough that we have no strong pre-ordained notions of what truth means.

In the first kind of area, mathematicians agree on what we want to say, we just don't know how to really say it.

In the second kind of area things are defined by taste. And taste is largely, "What makes for nicer theories?"

For example, the GCH implies that there are no infinite cardinals between the size of the set of integers and the size of the set of real numbers. I find that plausible and elegant -- but is it "true"? What would you look at to "see" whether or not that's true? The axiom of choice states that any set -- even an infinite set -- can be well-ordered. Again, I find this plausible, but is is "true"?

Well I'm the wrong person to comment on this since I'm a closet constructivist. :-)

Anyways if anyone finds this kind of topic interesting, then I strongly recommend Davis and Hersch's book The Mathematical Experience. It is readable for the layperson, but better than any other book that I know of conveys what it is like to be a mathematician. And while it can be read by a highschool student, people with higher degrees in mathematics still get a lot out of it. Perhaps the highest recommend that I can give is the number of mathematicians that I know who say that they became mathematicians because of The Mathematical Experience.

Cheers,
Ben

[Ben Tilly]

A few comments Re Godel.
As I understand it his proof applies only to systems with a finite set of axioms.
The reason inconsistency is a major problem is that in classical logic from a contradiction you can deduce any proposition you like. There are logicians that think that this aspect of classical logic is not correct, see paraconsistent logic and inconsistent mathematics.

A finite set of axioms is wrong. A finite description is true. That is, the axiom system may include a statement of the form, "All statements of form X are true." This is an infinite set of axioms. But the description is finite. So the theorem applies.

About logicians, I think it is too strong to say that there are logicians who think this aspect of classical logic is not correct. When reasoning about classical mathematics, any logician should accept classical reasoning. But they say that it is worth thinking about logic systems in which this assumption does not hold. And they point out cases where in the real world people have to do so.

Cheers,
Ben

[a_unique_person]

This is embarrassing, a thread I start that I can't understand. Carry on.

[switchtech]

A finite set of axioms is wrong. A finite description is true. That is, the axiom system may include a statement of the form, "All statements of form X are true." This is an infinite set of axioms. But the description is finite. So the theorem applies.

About logicians, I think it is too strong to say that there are logicians who think this aspect of classical logic is not correct. When reasoning about classical mathematics, any logician should accept classical reasoning. But they say that it is worth thinking about logic systems in which this assumption does not hold. And they point out cases where in the real world people have to do so.

Cheers,
Ben

Man, this thread started out great - but then it got into this bit about Mathematics and proof and provability and axioms and... I just completed a semester of Discreet Mathematics - my brain will explode if I have to dissect one more proof!.... :eek:

:D:D

[Mangafranga]

About logicians, I think it is too strong to say that there are logicians who think this aspect of classical logic is not correct. When reasoning about classical mathematics, any logician should accept classical reasoning. But they say that it is worth thinking about logic systems in which this assumption does not hold. And they point out cases where in the real world people have to do so.

Cheers,
Ben

I want to try and make things as clear as possible.
-there is the position that it is not possible that (p . ~p)
-there is the position that (p . ~p) does not imply everything
Do you think that all logicians hold one or both of these positions?

I have no idea what classical mathematics is.

[Ben Tilly]

I want to try and make things as clear as possible.
-there is the position that it is not possible that (p . ~p)
-there is the position that (p . ~p) does not imply everything
Do you think that all logicians hold one or both of these positions?

I have no idea what classical mathematics is.

The question you are asking betrays a fundamental misunderstanding. You always have to consider what system of logic you are working in.

There are systems of logic in which it is impossible for p and ~p to both hold. A well-known example is our usual way of doing mathematics.

There are other systems of logic in which it is possible for p and ~p to both hold without immediately implying every possible conclusion.

Most logicians are interested in the former. Some have studied the latter. Some of those who study the latter argue that the latter is more applicable to how people think about the world. But there is no question that the former is more descriptive of how mathematicians try to work.

Any logician who cannot keep it straight that the system of logic they are currently working in determines whether contradictions are allowed is hopelessly confused and can safely be ignored.

Hopefully that makes sense to you.

Cheers,
Ben

[Mangafranga]

The question you are asking betrays a fundamental misunderstanding. You always have to consider what system of logic you are working in.

I disagree in general with this. A system of logic does not just arise by magic, it should have philosophical justification. That is, its quite possible that logicians disagree about the systems of logic that are correct.

There are systems of logic in which it is impossible for p and ~p to both hold. A well-known example is our usual way of doing mathematics.

"Is (p . ~p) possible?" is a perfectly sensible question which can be asked outside of the context of a logical system.
Let "p" be "This sentence is false"
(It is true that "This sentence is false" and it is false that "This sentence is false") The question of if this is a true contradiction is a genuine philosophical issue.

There are other systems of logic in which it is possible for p and ~p to both hold without immediately implying every possible conclusion.

Most logicians are interested in the former. Some have studied the latter. Some of those who study the latter argue that the latter is more applicable to how people think about the world. But there is no question that the former is more descriptive of how mathematicians try to work.

Yes, part of the justification for paraconsistent logic is what you mentioned. But part of it is the belief by some logicians that true contradictions exist in the world. That it is the case that (p . ~p) in some instances. This is justification for disposing of the inference from (p .~p) to everything.

Only one reason for paraconsistent logic is to do with the way people think, so to then reject it as applying to mathematicians because this one justification doesn't hold is wrong.

The field of inconsistent mathematics would tend to disagree with you on the count that paraconsistent logic does not apply to mathematics.

Any logician who cannot keep it straight that the system of logic they are currently working in determines whether contradictions are allowed is hopelessly confused and can safely be ignored.


I would say that any logician who disregards philisophical justification for the system of logic they support can be ignored.

[69dodge]

"Is (p . ~p) possible?" is a perfectly sensible question which can be asked outside of the context of a logical system.I think I'd disagree with that.

You haven't described what sort of thing is supposed to be represented by the variable "p", nor what "~" means, nor what "." means, etc.

Without such a description, I can't answer the question, because I don't know what it's asking. With such a description, I'm working within the context of the logical system that it describes.

Let "p" be "This sentence is false"
(It is true that "This sentence is false" and it is false that "This sentence is false") The question of if this is a true contradiction is a genuine philosophical issue.The question seems sort of silly to me.

A sentence has to say something, in order to be true or false. We determine whether what it says, is or is not; and that's how we decide whether it's a true sentence or a false one. If it doesn't say anything, what is meant by calling it "true" or "false"?

To decide whether the sentence "the sun is shining" is true or false, we go outside and look up at the sky. The sentence says something about the sun, so we look at the sun to decide whether it's true or not.

"This sentence is false" appears to say something, but this is an illusion. It doesn't talk about anything real, like the sun; it just talks about the truth of some other sentence. So first we'd have to determine whether that sentence is true or not. But that sentence is the same sentence, and the process would go on forever, never arriving at anything which could be checked. Nothing is actually being asserted.

Yes, part of the justification for paraconsistent logic is what you mentioned. But part of it is the belief by some logicians that true contradictions exist in the world. That it is the case that (p . ~p) in some instances. This is justification for disposing of the inference from (p .~p) to everything.Can you give any examples, besides "this sentence is false"?

Do you think "this sentence is false" really talks about something that exists in the world, the way "the sun is shining" talks about something that exists in the world, namely the sun? I think it's just wordplay.

Not everything that can be said means anything. Shall we worry about whether "asd;lfkj" is a true sentence or a false one, too?

[Mangafranga]

I think I'd disagree with that.

You haven't described what sort of thing is supposed to be represented by the variable "p", nor what "~" means, nor what "." means, etc.

Perhaps I should not have used notation. "p"=df a statement which can be assed for truth value. "~"=df negation. "."=df conjuction

Without such a description, I can't answer the question, because I don't know what it's asking. With such a description, I'm working within the context of the logical system that it describes.

Perhaps I should not have used notation. "p"=df a statement which can be assed for truth value. "~"=df negation. "."=df conjuction. We can junk the notation and abstraction if you like. Is it possible that there is an apple in my hand right now and that there is not an apple in my hand right now?

The question seems sort of silly to me.

A sentence has to say something, in order to be true or false. We determine whether what it says, is or is not; and that's how we decide whether it's a true sentence or a false one. If it doesn't say anything, what is meant by calling it "true" or "false"?

To decide whether the sentence "the sun is shining" is true or false, we go outside and look up at the sky. The sentence says something about the sun, so we look at the sun to decide whether it's true or not.

"This sentence is false" appears to say something, but this is an illusion. It doesn't talk about anything real, like the sun; it just talks about the truth of some other sentence. So first we'd have to determine whether that sentence is true or not. But that sentence is the same sentence, and the process would go on forever, never arriving at anything which could be checked. Nothing is actually being asserted.

This is a position that some philosophers hold. Others, like Graham Priest, hold that this is a true contradiction.

Can you give any examples, besides "this sentence is false"?

Do you think "this sentence is false" really talks about something that exists in the world, the way "the sun is shining" talks about something that exists in the world, namely the sun? I think it's just wordplay.

Not everything that can be said means anything. Shall we worry about whether "asd;lfkj" is a true sentence or a false one, too?

Graham Priest (ex mathematician btw) "In Contradiction" argues that motion is best acounted for by inconsistency. Chris Mortesen "Paradoxes inside and outside language" argues for visual paradoxes.

My point has not been to personally argue for or against true contradictions. My point has been to argue that there exist philosophers who do so, and that this argument will motivate a system of logic.

[Ben Tilly]

I disagree in general with this. A system of logic does not just arise by magic, it should have philosophical justification. That is, its quite possible that logicians disagree about the systems of logic that are correct.

Logicians may disagree about what system of logic we should use. That can be an interesting philosophical question for those who are inclined. But logicians need to be able to say, "In this system of logic, here is what follows."

Therefore no person is worthy of the name "logician" who can't understand and apply traditional logic. And the window of opportunity to get mathematicians as a whole to reconsider their approach closed a few decades ago. If you want to re-open it you'd need to do something drastic. Proving that ZFC is inconsistent would suffice. Failing that you're simply not going to get mathematicians to reconsider how they operate.

BTW it may interest you to know the philosophical grounds that mathematicians had for the approach that they settled for. Those grounds were, "This makes it easier to do mathematics, and alternatives such as constructivism are no more likely to work than this." (Yes, the second part of that is a theorem. By, as you might guess, Gödel.) You'll note that, "This makes sense" and "This accords with naive notions" do not appear in those criteria...

"Is (p . ~p) possible?" is a perfectly sensible question which can be asked outside of the context of a logical system.
Let "p" be "This sentence is false"
(It is true that "This sentence is false" and it is false that "This sentence is false") The question of if this is a true contradiction is a genuine philosophical issue.

This is certainly an important example. It is, for instance, why Cantor's original approach to set theory fell apart. And there are many approaches to dealing with it.

The approach that most of mathematics uses is to use logical systems that try to make that statement impossible to state. To the best of our knowledge, they succeeded. To the best of my knowledge, no alternative has been found that is likely to be more consistent.

Yes, part of the justification for paraconsistent logic is what you mentioned. But part of it is the belief by some logicians that true contradictions exist in the world. That it is the case that (p . ~p) in some instances. This is justification for disposing of the inference from (p .~p) to everything.

Mathematics is not the real world.

Only one reason for paraconsistent logic is to do with the way people think, so to then reject it as applying to mathematicians because this one justification doesn't hold is wrong.

Sorry, but you aren't making sense here. I indicated that mathematicians try to work within a consistent logic. There are exceptions, but they are few and far between. Therefore paraconsistent logic simply is irrelevant to what most mathematicians are trying to do.

Paraconsistent logic might be applicable to describing how those mathematicians actually work. (A dirty secret of mathematics is that mathematicians don't actually believe all of the theorems that are published, they know that there are a certain number of errors in the reasoning...) However it is irrelevant to what most mathematicians are actually doing.

The field of inconsistent mathematics would tend to disagree with you on the count that paraconsistent logic does not apply to mathematics.

It applies to a rather small fraction of mathematics.

I'd wager that most PhD's in mathematics have never heard of paraconsistent logic, let alone learned it. By contrast all logicians who are involved with paraconsistent logic have had to learn classical logic. And most of those who are at universities have to accept it enough to be able to teach it.

I would say that any logician who disregards philisophical justification for the system of logic they support can be ignored.

There are many logicians working in mathematics departments who can be ignored by your criteria. Very few to none by mine.

Cheers,
Ben

[Mangafranga]

I'm not sure that we haven't started talking at cross purposes. What has confounded me is that you seem to be saying that you cannot discuss logic without using a formalized system of logic, are you in fact saying this?

[lupus_in_fabula]

The concept of time is a rather handy tool, at times :)

[DevilsAdvocate]

The scientific method. Sans comparison.I agree. Language/writing, tools/wheels, fire, agriculture, can only get you so far. Even if humans didn’t have any of those things, using the scientific method to resolve problems or accomplish goals could have resulted in any of those other discoveries. The wheel has been reinvented. All of those other discoveries were achieved by a rudimentary use of the scientific method-a trial and error method. The scientific method cannot be compared.

[Ben Tilly]

I'm not sure that we haven't started talking at cross purposes. What has confounded me is that you seem to be saying that you cannot discuss logic without using a formalized system of logic, are you in fact saying this?

You finally noticed?

I would phrase it as, "The rules of logic available to you depend on the system of logic you are operating within." Trying to discuss them outside of a system of logic (be that implicit or explicit) is like asking whether the Empire State Building is near or far from you without being willing to say where you physically are.

Cheers,
Ben

[69dodge]

Perhaps I should not have used notation. "p"=df a statement which can be assed for truth value. "~"=df negation. "."=df conjuction. We can junk the notation and abstraction if you like. Is it possible that there is an apple in my hand right now and that there is not an apple in my hand right now?Yes, "~" means negation, but that definition doesn't help until we decide what "negation" means.

It's not a question of fact, independent of any system of formal logic, whether p and ~p might both be true; it's a question of whether we want, in our system of formal logic, to call two sentences "p" and "~p" if they might both be true.

Why is "there is not an apple in my hand" the negation of "there is an apple in my hand"? Just because the first sentence consists of the same words as the second except for the insertion of the word "not"? That seems a very superficial way of looking at things. We should examine the meanings of the two sentences, not just the way these meanings happen to be expressed in English.

You have two hands. Suppose one is empty while the other holds an apple. Do you mean for the sentence "there is an apple in my hand" to describe this situation? What about the sentence "there is not an apple in my hand"? It could plausibly be argued that both apply, but in that case I wouldn't call the second the negation of the first. Would you? Why?

The point of formalizing logic is, in my opinion, to help us be as clear and precise as we can be, not simply to mimic in a different symbolism the ambiguities of natural languages like English.

(I'm not sure if you intended to hint at the ambiguity of "my hand". Maybe "right now" was the important phrase? In any event, all you need to do is make it sufficiently clear what you mean by a sentence. "Making it clear what you mean by a sentence" just means saying under what circumstances you intend it to be considered true. Then, under those circumstances, it's true but not false; and under all others, it's false but not true. What could possibly be meant by the assertion that it's both true and false? "False" means not true!)

Graham Priest (ex mathematician btw) "In Contradiction" argues that motion is best acounted for by inconsistency. Chris Mortesen "Paradoxes inside and outside language" argues for visual paradoxes.Thanks for the references.

Sounds weird, I must say. But I guess I really should read them before denouncing them as utter nonsense. :D

[tracer]

Bah. Amateurs, all.

The gratest discovery of all time was Nicholas LeBlanc's 1794 discovery of a way to (inexpensively) make Sodium Carbonate out of Sodium Chloride.

This led directly to the wide-spread availability of ... SOAP !!!

Try imagining life in the modern world without soap. You can't, can you?

[Mangafranga]

You finally noticed?

I would phrase it as, "The rules of logic available to you depend on the system of logic you are operating within." Trying to discuss them outside of a system of logic (be that implicit or explicit) is like asking whether the Empire State Building is near or far from you without being willing to say where you physically are.

Cheers,
Ben

I disagree. Logic is the study of reasoning. It tells us wether or not we can validly infer a proposition on the basis of the premises. It tells us if we can say that the conclusion must be true on the basis of the premsies.

Socrates is a man
All men are mortal
Therefore Socrates is mortal

This is a valid argument

Socrates is mortal
All men are mortal
Therefore Socrates is a man

This is an invalid argument. It is not invalid because of the rules of a particular logic.

I suggest you read the entries on Stanford Encyclopedia of Philosophy on paraconsistent logic and dialethism. They support my account.

[DanishDynamite]

"The Greatest Discovery", unless defined in specifics, is unaswerable.

However, on the face of the lack of a proper definition, I, a physicist in spirit but an engineer in reality, would go with a suggestion I saw somewhere upstream, namely the invention of agriculture.

[Mangafranga]

Yes, "~" means negation, but that definition doesn't help until we decide what "negation" means.

It's not a question of fact, independent of any system of formal logic, whether p and ~p might both be true; it's a question of whether we want, in our system of formal logic, to call two sentences "p" and "~p" if they might both be true.

Why is "there is not an apple in my hand" the negation of "there is an apple in my hand"? Just because the first sentence consists of the same words as the second except for the insertion of the word "not"? That seems a very superficial way of looking at things. We should examine the meanings of the two sentences, not just the way these meanings happen to be expressed in English.

I left the ambiguity in there to be ignored.
The negation of "It is the case that there exists at least one and not more than one James Randi" is "It is not the case that there exists at least one and not more than one James Randi". The conjuction of them is "It is the case that there exists at least one and not more than one James Randi and it is not the case that there exists at least one and not more than one James Randi".
Is it possible that it is the case that there exists at least one and not more than one James Randi and it is (also) not the case that there exists at least one and not more than one James Randi?
Is this a question of fact, independent of any system of formal logic, and for that matter independent of how the world actually is (do you need to do physical investigation into the world to find the answer) ?

Thanks for the references.

Sounds weird, I must say. But I guess I really should read them before denouncing them as utter nonsense.

They are both Professors writing in their area of expertese. I misspelled Mortensen, whose article is avaliable online if you have accsess to journal databases.

[Ben Tilly]

I disagree. Logic is the study of reasoning. It tells us wether or not we can validly infer a proposition on the basis of the premises. It tells us if we can say that the conclusion must be true on the basis of the premsies.

Socrates is a man
All men are mortal
Therefore Socrates is mortal

This is a valid argument

Aristotle would be proud of you.

Socrates is mortal
All men are mortal
Therefore Socrates is a man

This is an invalid argument. It is not invalid because of the rules of a particular logic.

Aristotle would give you a gold star.

Of course Aristotle would flunk you instantly once he found out that you were going from this to saying that contradictions are OK.

I suggest you read the entries on Stanford Encyclopedia of Philosophy on paraconsistent logic and dialethism. They support my account.

I just read them. I find in the section on paraconsistent logic statements like, There are many different paraconsistent logics. Most of them can be defined in terms of a semantics which allows both A and ~A to hold in an interpretation.

Notice something? Perhaps if I highlight a key word it might help. There are many different paraconsistent logics.

What they are referring to with the word "logics" is what I meant by "systems of logic". Which means that it certainly doesn't support your account.

BTW an aside to everyone else. I have the feeling that we're going round and round on this topic but overall we're going nowhere. I may be wrong, but I really get the strong impression that Mangafranga is someone who has read a couple of books that he only half-understood which he has become very convinced by. And he's therefore trying to convince the rest of the world. But he keeps on missing basic points, and misstating others, and I'm getting tired of this discussion. Therefore unless someone specifically requests otherwise, I'm going to drop this thread.


Cheers,
Ben

[Mangafranga]

Of course Aristotle would flunk you instantly once he found out that you were going from this to saying that contradictions are OK.

I never said that contradictions are OK, I said some philosophers held this position. From Stanford Encyclopedia of Philosophy, Dialetheism

" A dialetheia is a true contradiction, a statement, A, such that both it and its negation, missing text, please informA, are true. Hence, dialeth(e)ism is the view that there are true contradictions."

But I wonder how Aristotle would demonstrate that this imagined argument was wrong? Considering that logic is used to analyise arguments, he could translate this imagined argument to symoblism and analyise the structure of the argument. But according to you there is no objective mesure of logic, and so then this would be unjustified.

He could also just argue against the statement that contradictions are possible. Which he did, from Stanford Encyclopedia of Philosophy, Contradiction

" The twin foundations of Aristotle's logic are the law of non-contradiction (LNC) (also called the law of contradiction) and the law of excluded middle (LEM). In Metaphysics Book Gamma, LNC—“the most certain of all principles”—is defined as follows:

It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect, and all other specifications that might be made, let them be added to meet local objections (1005b19-23)."

"We can translate the Aristotelian language, with some loss of faithfulness, into the standard modern versions in (4a,b) respectively, ignoring the understood modal and temporal modifications:

(4a) LNC: ¬(Φ & ¬Φ)
(4b) LEM: Φ v ¬Φ"

So to make my position as clear as possible. I contend that logic studies reasoning. Reasoning is good or bad in and of itself. Any logic which allowed inferences of the type

Socrates is mortal
All men are mortal
Therefore Socrates is a man

would be wrong. Modus ponens on the other hand is a correct argument form, and any system of logic which rejected it would be wrong.

I also claim that there are philosophers who believe that contradictions are possible. From Stanford Encyclopedia of Philosophy, Dialetheism

"A dialetheia is a true contradiction, a statement, A, such that both it and its negation, missing text, please informA, are true. Hence, dialeth(e)ism is the view that there are true contradictions. Dialetheism opposes the so-called Law of Non-Contradiction (LNC)..."

I claim that if this view were correct it would justify the rejection of Ex Contradictione Quodlibet (from a contradicition everything validly follows). From Stanford Encyclopedia of Philosophy, Paraconsistent Logic

"If there are dialetheias then some inferences of the form {A , ~A} models B must fail. For only true conclusions follow validly from the true premises. Hence logic has to be paraconsistent."

I claim that the rejection of ECQ would justify saying that inconsistency is not fatal. From Stanford Encyclopedia of Philosophy, Inconsistent Mathematics

"Hence, a number of people including da Costa (1974), Brady (1971), Priest, Routley, and Norman (1989), considered it preferable to retain the full power of the natural abstraction principle (every predicate determines a set), and tolerate a degree of inconsistency in set theory. This requires, of course, that one dispense with the logical principle ex contradictione quodlibet (ECQ) (from a contradiction every proposition may be deduced), as well as any principle which leads to it, such as disjunctive syllogism (DS) (from A-or-B and not-A deduce B). But considerable debate, in Burgess (1981) and Mortensen (1983), made it clear that dispensing with ECQ and DS was not so counter-intuitive, especially when a plausible story emerged about the special conditions under which they continue to hold."

BTW an aside to everyone else. I have the feeling that we're going round and round on this topic but overall we're going nowhere. I may be wrong, but I really get the strong impression that Mangafranga is someone who has read a couple of books that he only half-understood which he has become very convinced by. And he's therefore trying to convince the rest of the world. But he keeps on missing basic points, and misstating others, and I'm getting tired of this discussion. Therefore unless someone specifically requests otherwise, I'm going to drop this thread.

I'm happy to leave the thread, but I'll do it without insulting you.