Discussion Forum > A test for clear thinking
(Hint: if your initial reaction is "What a silly question, it's obviously NNN" or some form thereof -- you are not thinking clearly.)
February 10, 2014 at 17:31 |
Seraphim
Seraphim
Love this question. I would say mathematical laws are discovered, but mathematical notation and such are a language like any other and thus invented.
February 10, 2014 at 17:48 |
Vegheadjones
Vegheadjones
Since my reaction was immediately "What a stupid question, it's obviously a discovery", I guess the correct answer must be that it's an invention.
The question is really "What's the difference between an invention and a discovery?" and when one starts thinking in that way it becomes apparent that discoveries are part of inventions. For instance if you set out to invent a better mousetrap, you might do so by discovering that mice are attracted by warm, deep colours and low frequency noise. (I've no idea if they really are or not). Or possibly you might discover the exact tension that is required on a spring to trip a mousetrap for an adult mouse. Or you might discover that mice are particularly fond of ripe gorgonzola. So making discoveries is an essential part of inventing.
In the same way, making discoveries is an essential part of mathematics, but the whole is an invention.
The question is really "What's the difference between an invention and a discovery?" and when one starts thinking in that way it becomes apparent that discoveries are part of inventions. For instance if you set out to invent a better mousetrap, you might do so by discovering that mice are attracted by warm, deep colours and low frequency noise. (I've no idea if they really are or not). Or possibly you might discover the exact tension that is required on a spring to trip a mousetrap for an adult mouse. Or you might discover that mice are particularly fond of ripe gorgonzola. So making discoveries is an essential part of inventing.
In the same way, making discoveries is an essential part of mathematics, but the whole is an invention.
February 10, 2014 at 18:30 |
Mark Forster
Mark Forster
Before reading the comments above, I thought about it a bit, and said "Both." Vegheadjones's comment succinctly explains why.
February 10, 2014 at 18:53 |
ubi
ubi
I agree with Vegheadjones.
February 11, 2014 at 14:19 |
Cricket
Cricket
So what's your answer, Seraphim?
February 25, 2014 at 22:37 |
Mark Forster
Mark Forster
Speaking as one with both a degree in, and a passion for mathematics, I can say it clearly involves both. A student would invent a system of study, and then will investigate and discover its properties. In the process he may invent new methods of investigation and proceed to prove (discover) the scope in which it applies. In general I might call the axioms and structures inventions, and theorems discoveries.
February 26, 2014 at 2:15 |
Alan Baljeu
Alan Baljeu
OK, you can google now. :-) http://www.google.com/search?q=mathematics+discovery+or+invention%3F
And you will find that some of the greatest thinkers have come to different conclusions on this topic.
A pretty good summary: http://www.huffingtonpost.com/derek-abbott/is-mathematics-invented-o_b_3895622.html
A fascinating interview on this topic:
Part 1 - http://www.idthefuture.com/2014/01/david_berlinski_the_nature_of_.html
Part 2 - http://www.idthefuture.com/2014/01/david_berlinski_the_nature_of.html
An interesting essay examining whether mathematics is a form of philosophical Platonic idealism can be found here: http://www.ems-ph.org/journals/newsletter/pdf/2008-06-68.pdf (page 19 of the PDF)
My personal view is that we must make a clear distinction between reality and the semiotic forms by which we conceive of reality. We often make propositional statements and declare them to be "true" without having a clear idea of what such an assertion even means. Such a statement presupposes an actual correspondence between the statement and its analog in the "real world". However, such presuppositions often fail to acknowledge that the statement is the product of a semiotic system of language with an ambiguous relationship to the "real world". The problem is compounded when different people do not agree on the meaning of words -- in other words, they propose (or assume) a variant correspondence between the statement and the reality it is intended to represent.
This problem would seem to be diminished in mathematics, in comparison to natural human language. However, it is still a fundamentally intractable problem, as demonstrated by Godel's Incompleteness Theorem. The theorem proves that any arithmetic system is either "complete" (describing fully the underlying reality), or consistent (i.e., not self-contradictory). It CANNOT be both.
In other words, if an arithmetical semiotic system is consistent, it cannot be complete.
And if such a system is complete, it cannot be consistent - it must be self-contradictory.
This is quite a conundrum, because it proves that mathematics cannot completely describe the reality of numbers and their properties. It does not merely suggest this conclusion, but proves it. It also proves that any arithmetic formal system that claims to be complete must necessarily be self-contradictory.
http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems
This theorem has been extended to formal systems, semiotics, and science in general (Tarski's undefinability theorem): a complete 'unified field theory' or "description of everything" will necessarily be self-contradictory. (Mathematicians and scientists need not fear losing their jobs because of all the answers having been discovered. :-)
http://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem
So: back to the original question. If by the term "mathematics" we mean the semiotic system that attempts to describe the reality of numbers and their properties, then I would say it is an invention - perhaps an invention with uniquely interesting properties, but an invention nonetheless. But if we mean the underlying objective reality itself, and our ability to understand some of its properties, then I would say it is a discovery. It is very difficult (impossible?) to separate the two ideas: so I would agree with Alan, that it involves both. But I disagree with Alan when he says it CLEARLY involves both, because many intelligent and knowledgeable people have found it to be CLEARLY one or the other. :-)
And you will find that some of the greatest thinkers have come to different conclusions on this topic.
A pretty good summary: http://www.huffingtonpost.com/derek-abbott/is-mathematics-invented-o_b_3895622.html
A fascinating interview on this topic:
Part 1 - http://www.idthefuture.com/2014/01/david_berlinski_the_nature_of_.html
Part 2 - http://www.idthefuture.com/2014/01/david_berlinski_the_nature_of.html
An interesting essay examining whether mathematics is a form of philosophical Platonic idealism can be found here: http://www.ems-ph.org/journals/newsletter/pdf/2008-06-68.pdf (page 19 of the PDF)
My personal view is that we must make a clear distinction between reality and the semiotic forms by which we conceive of reality. We often make propositional statements and declare them to be "true" without having a clear idea of what such an assertion even means. Such a statement presupposes an actual correspondence between the statement and its analog in the "real world". However, such presuppositions often fail to acknowledge that the statement is the product of a semiotic system of language with an ambiguous relationship to the "real world". The problem is compounded when different people do not agree on the meaning of words -- in other words, they propose (or assume) a variant correspondence between the statement and the reality it is intended to represent.
This problem would seem to be diminished in mathematics, in comparison to natural human language. However, it is still a fundamentally intractable problem, as demonstrated by Godel's Incompleteness Theorem. The theorem proves that any arithmetic system is either "complete" (describing fully the underlying reality), or consistent (i.e., not self-contradictory). It CANNOT be both.
In other words, if an arithmetical semiotic system is consistent, it cannot be complete.
And if such a system is complete, it cannot be consistent - it must be self-contradictory.
This is quite a conundrum, because it proves that mathematics cannot completely describe the reality of numbers and their properties. It does not merely suggest this conclusion, but proves it. It also proves that any arithmetic formal system that claims to be complete must necessarily be self-contradictory.
http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems
This theorem has been extended to formal systems, semiotics, and science in general (Tarski's undefinability theorem): a complete 'unified field theory' or "description of everything" will necessarily be self-contradictory. (Mathematicians and scientists need not fear losing their jobs because of all the answers having been discovered. :-)
http://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem
So: back to the original question. If by the term "mathematics" we mean the semiotic system that attempts to describe the reality of numbers and their properties, then I would say it is an invention - perhaps an invention with uniquely interesting properties, but an invention nonetheless. But if we mean the underlying objective reality itself, and our ability to understand some of its properties, then I would say it is a discovery. It is very difficult (impossible?) to separate the two ideas: so I would agree with Alan, that it involves both. But I disagree with Alan when he says it CLEARLY involves both, because many intelligent and knowledgeable people have found it to be CLEARLY one or the other. :-)
February 26, 2014 at 6:46 |
Seraphim
Seraphim
I'll go with engineering. What's the easiest model that will give me an accurate-enough prediction? Is there an easier one I can use for the initial calculations? Will a more accurate model be worth the investment? I value mathematicians and theorists who keep making more accurate theories (we'd not be past the Industrial Revolution without them), and enjoy learning them (well, back when I could actually do basic partial differential equations), but working engineers are more concerned with staying within the limits of the models than expanding them.
February 26, 2014 at 16:03 |
Cricket
Cricket
From the article ( http://www.huffingtonpost.com/derek-abbott/is-mathematics-invented-o_b_3895622.html )
<quote>
Amongst mathematicians and scientists there is no consensus on this fascinating question. The various types of responses to Einstein's conundrum include:
1) Math is innate. The reason mathematics is the natural language of science, is that the universe is underpinned by the same order. The structures of mathematics are intrinsic to nature. Moreover, if the universe disappeared tomorrow, our eternal mathematical truths would still exist. It is up to us to discover mathematics and its workings--this will then assist us in building models that will give us predictive power and understanding of the physical phenomena we seek to control. This rather romantic position is what I loosely call mathematical Platonism.
2) Math is a human construct. The only reason mathematics is admirably suited describing the physical world is that we invented it to do just that. It is a product of the human mind and we make mathematics up as we go along to suit our purposes. If the universe disappeared, there would be no mathematics in the same way that there would be no football, tennis, chess or any other set of rules with relational structures that we contrived. Mathematics is not discovered, it is invented. This is the non-Platonist position.
3) Math is not so successful. Those that marvel at the ubiquity of mathematical applications have perhaps been seduced by an overstatement of their successes. Analytical mathematical equations only approximately describe the real world, and even then only describe a limited subset of all the phenomena around us. We tend to focus on those physical problems for which we find a way to apply mathematics, so overemphasis on these successes is a form of "cherry picking." This is the realist position.
4) Keep calm and carry on. What matters is that mathematics produces results. Save the hot air for philosophers. This is called the "shut up and calculate" position.
</quote>
Sounds like you are in category #4, Cricket. :-)
<quote>
Amongst mathematicians and scientists there is no consensus on this fascinating question. The various types of responses to Einstein's conundrum include:
1) Math is innate. The reason mathematics is the natural language of science, is that the universe is underpinned by the same order. The structures of mathematics are intrinsic to nature. Moreover, if the universe disappeared tomorrow, our eternal mathematical truths would still exist. It is up to us to discover mathematics and its workings--this will then assist us in building models that will give us predictive power and understanding of the physical phenomena we seek to control. This rather romantic position is what I loosely call mathematical Platonism.
2) Math is a human construct. The only reason mathematics is admirably suited describing the physical world is that we invented it to do just that. It is a product of the human mind and we make mathematics up as we go along to suit our purposes. If the universe disappeared, there would be no mathematics in the same way that there would be no football, tennis, chess or any other set of rules with relational structures that we contrived. Mathematics is not discovered, it is invented. This is the non-Platonist position.
3) Math is not so successful. Those that marvel at the ubiquity of mathematical applications have perhaps been seduced by an overstatement of their successes. Analytical mathematical equations only approximately describe the real world, and even then only describe a limited subset of all the phenomena around us. We tend to focus on those physical problems for which we find a way to apply mathematics, so overemphasis on these successes is a form of "cherry picking." This is the realist position.
4) Keep calm and carry on. What matters is that mathematics produces results. Save the hot air for philosophers. This is called the "shut up and calculate" position.
</quote>
Sounds like you are in category #4, Cricket. :-)
February 26, 2014 at 22:17 |
Seraphim
Seraphim
Seraphim:
<< So: back to the original question. If by the term "mathematics" we mean the semiotic system that attempts to describe the reality of numbers and their properties, then I would say it is an invention - perhaps an invention with uniquely interesting properties, but an invention nonetheless. But if we mean the underlying objective reality itself, and our ability to understand some of its properties, then I would say it is a discovery. It is very difficult (impossible?) to separate the two ideas: so I would agree with Alan, that it involves both. >>
Or, to put it perhaps a bit more simply, you agree with what I said in the fourth post in this thread.
<< So: back to the original question. If by the term "mathematics" we mean the semiotic system that attempts to describe the reality of numbers and their properties, then I would say it is an invention - perhaps an invention with uniquely interesting properties, but an invention nonetheless. But if we mean the underlying objective reality itself, and our ability to understand some of its properties, then I would say it is a discovery. It is very difficult (impossible?) to separate the two ideas: so I would agree with Alan, that it involves both. >>
Or, to put it perhaps a bit more simply, you agree with what I said in the fourth post in this thread.
February 27, 2014 at 0:50 |
Mark Forster
Mark Forster
Consider the game of Chess... Chess itself is an invention: the pieces, the board, & crucially the arbitrary rules that govern play. There are 10^120 possible games of Chess. However, all playable games of Chess are potentially contained in the basic parameters and rules of the game. Therefore each new move, tactic, & strategy could be said to be ‘discovery’, rather than ‘invention’, within the game.
It could be argued that Mathematics is an 'invention' because the entire construct is a formal system, created by human intellect. It is not 'discovered' in the way that a physical law like gravitation is: by observation, induction, hypothesis formation, & empirical testing.
However, I would argue that Mathematics involves more discovery than invention.
First: Mathematics & Logic both involve discovering the underlying mechanisms of their respective but related disciplines, i.e. a priori principles & formal systems, built on:
(i) fundamental concepts such as number, space, logic etc;
(ii) axioms & postulates
(iii) controlled & formal rules of inference, manipulation & derivation that ensure that inferences made within the system are valid and that truth is maintained;
(iv) symbolic notation.
Second: new mathematical principles, propositions, theorems etc. are derived by deductive inference within the systems, or additions/modifications to the system. In principle, these truths & derivations are intrinsically present within the formal system, and thus it could argued that mathematics is a 'discovery' process because it involves extrapolating from them.
I'd argue therefore that Mathematics is more 'discovery' than 'invention'.
It could be argued that Mathematics is an 'invention' because the entire construct is a formal system, created by human intellect. It is not 'discovered' in the way that a physical law like gravitation is: by observation, induction, hypothesis formation, & empirical testing.
However, I would argue that Mathematics involves more discovery than invention.
First: Mathematics & Logic both involve discovering the underlying mechanisms of their respective but related disciplines, i.e. a priori principles & formal systems, built on:
(i) fundamental concepts such as number, space, logic etc;
(ii) axioms & postulates
(iii) controlled & formal rules of inference, manipulation & derivation that ensure that inferences made within the system are valid and that truth is maintained;
(iv) symbolic notation.
Second: new mathematical principles, propositions, theorems etc. are derived by deductive inference within the systems, or additions/modifications to the system. In principle, these truths & derivations are intrinsically present within the formal system, and thus it could argued that mathematics is a 'discovery' process because it involves extrapolating from them.
I'd argue therefore that Mathematics is more 'discovery' than 'invention'.
February 27, 2014 at 11:21 |
James Precious
James Precious
One could argue that there is no difference between a discovery and an invention, since the word "invention" comes from the Latin "invenire" which means, you've guessed it, "to discover".
February 27, 2014 at 17:02 |
Mark Forster
Mark Forster
This is exactly why the question as originally posed is a good test for clear thinking. To get clarity here, the first step is to define what is meant by all three terms - mathematics, invention, and discovery. There are many hidden assumptions in the way people interpret all three terms.
In my last post, I tried to elucidate what I mean by "mathematics" but didn't go into what I mean by "invention" and "discovery". Perhaps the way the Huffington article frames the discussion is more useful: "math is a human construct" (invention) vs. "math is innate" (discovery). But perhaps not. Different people may think about these questions in different ways. I think that's why it's so hard to arrive at consensus and clarity.
In my last post, I tried to elucidate what I mean by "mathematics" but didn't go into what I mean by "invention" and "discovery". Perhaps the way the Huffington article frames the discussion is more useful: "math is a human construct" (invention) vs. "math is innate" (discovery). But perhaps not. Different people may think about these questions in different ways. I think that's why it's so hard to arrive at consensus and clarity.
February 27, 2014 at 17:22 |
Seraphim
Seraphim
Seraphim:
<< Different people may think about these questions in different ways. I think that's why it's so hard to arrive at consensus and clarity. >>
If it's possible for intelligent people to have different views about whether mathematics is an invention or a discovery, does that fact in itself not suggest that mathematics is an invention? It's very difficult to disagree that a discovery is a discovery - we don't have arguments about whether radioactivity or black holes are discoveries or inventions.
<< Different people may think about these questions in different ways. I think that's why it's so hard to arrive at consensus and clarity. >>
If it's possible for intelligent people to have different views about whether mathematics is an invention or a discovery, does that fact in itself not suggest that mathematics is an invention? It's very difficult to disagree that a discovery is a discovery - we don't have arguments about whether radioactivity or black holes are discoveries or inventions.
February 27, 2014 at 18:48 |
Mark Forster
Mark Forster
You could argue that Mark, but I think the disagreement comes from factors that dodge your logic. Some argue that God invented truth, including math, and all we do is discover the rules God established. Finding a patent at the patent office is merely discovering another's invention.
In the end, there is as you suggested earlier very little difference between the words. To invent something is generally a process of trying various ideas with an object of discovering an idea that works for your purpose. And if in the process you accidentally discover something useful in another way, you might file your discovery at the patent office as your new invention.
In the end, there is as you suggested earlier very little difference between the words. To invent something is generally a process of trying various ideas with an object of discovering an idea that works for your purpose. And if in the process you accidentally discover something useful in another way, you might file your discovery at the patent office as your new invention.
February 27, 2014 at 19:20 |
Alan Baljeu
Alan Baljeu
Alan:
<< Some argue that God invented truth, including math, and all we do is discover the rules God established. >>
But others would argue that the universe works perfectly well without using mathematics. And you probably drive your car perfectly well without using mathematics. Mathematics is merely a mental construct we have created in order to understand and manipulate certain aspects of reality.
<< Some argue that God invented truth, including math, and all we do is discover the rules God established. >>
But others would argue that the universe works perfectly well without using mathematics. And you probably drive your car perfectly well without using mathematics. Mathematics is merely a mental construct we have created in order to understand and manipulate certain aspects of reality.
February 27, 2014 at 22:34 |
Mark Forster
Mark Forster
Marc Forster said:
'One could argue that there is no difference between a discovery and an invention, since the word "invention" comes from the Latin "invenire" which means, you've guessed it, "to discover".
While Humpty Dumpty said: 'When I make a word do a lot of work like that I always pay it extra.'
Personally I think Mark captured "the reality" with "Mathematics is merely a mental construct we have created in order to understand and manipulate certain aspects of reality." and would just add .... so are all our languages (of which all the branches of mathematics are special instances).
'One could argue that there is no difference between a discovery and an invention, since the word "invention" comes from the Latin "invenire" which means, you've guessed it, "to discover".
While Humpty Dumpty said: 'When I make a word do a lot of work like that I always pay it extra.'
Personally I think Mark captured "the reality" with "Mathematics is merely a mental construct we have created in order to understand and manipulate certain aspects of reality." and would just add .... so are all our languages (of which all the branches of mathematics are special instances).
February 27, 2014 at 22:48 |
Bricoleur
Bricoleur
Most of my day, definitely category 4, but I enjoy excursions into the other categories, and value the work others do in the other categories.
The chess example. I was firmly in "we discover natural laws" until I read that. Maybe inventing and discovering are two ends of a spectrum? Or two independent acts that often occur together.
Still struggling with Godel's theorem. A complete and accurate description has to be as self-consistent as the system it's describing. One of those convoluted things that will ultimately help someone create something that will help someone else create something that someone else will use to directly affect my life, like magnetic resonance and MRIs.
The chess example. I was firmly in "we discover natural laws" until I read that. Maybe inventing and discovering are two ends of a spectrum? Or two independent acts that often occur together.
Still struggling with Godel's theorem. A complete and accurate description has to be as self-consistent as the system it's describing. One of those convoluted things that will ultimately help someone create something that will help someone else create something that someone else will use to directly affect my life, like magnetic resonance and MRIs.
March 1, 2014 at 20:30 |
Cricket
Cricket
The book "Godel, Escher, Bach: An eternal golden braid" by Douglas Hofstadter has an explanation of Godel's theorem that I found very convincing. Too long ago to reproduce it though, sorry, but it has something to do with the statement "All Kretenzers are liars" made by a Kretenzer. If a complete system has to be able to include statements like this it's inconsistent with itself, but if you exclude statements like this from the system it's incomplete.
March 3, 2014 at 11:46 |
Nicole
Nicole
“If you wish to converse with me,define your terms.” Voltaire.
Clear definitions usually do help with questions of concept, like this.
Clear definitions usually do help with questions of concept, like this.
March 3, 2014 at 20:29 |
James Precious
James Precious
Nicole - That is a very thought-provoking book - thanks for bringing it up. Can you give a page reference for his discussion of Godel's theorem? Thanks!!
March 3, 2014 at 20:33 |
Seraphim
Seraphim
Seraphim, I'll have to check the book at home, I'll try to look it up later this week. Yes, it's a very thought-provoking book. I'm not sure how relevant the discussion about DNA-encoding still is, research in that area has proceeded at incredable speeds, but the link between DNA-encoding as a 'complete' system and Gödel's theorem was interesting.
March 4, 2014 at 11:50 |
Nicole
Nicole
Definitions are useful for a conversation, yes.
Equally useful is the conversation to create the definitions, if the goal is to look at the options rather than for everyone to agree on the definitions. I've learned many different ways of looking at things and many subtle differences in thought through discussing definitions, even though when I don't change how I use the word.
Equally useful is the conversation to create the definitions, if the goal is to look at the options rather than for everyone to agree on the definitions. I've learned many different ways of looking at things and many subtle differences in thought through discussing definitions, even though when I don't change how I use the word.
March 4, 2014 at 20:02 |
Cricket
Cricket
Seraphim:
The "proof" of Gödel's theorem in Douglas Hofstadter's book is on page 271-272 in my edition (under the subheading: "G: a string which talks about itself in code"). The entire text up to that point explains number theory in layman's terms, and what Gödel's reasoning was in developing his theorem, so it's probably difficult to understand those pages if you haven't read what leads up to that point.
Part of what made GEB an interesting book for me to read was that it mixes computer and number theory with a description of how DNA encoding works, and with the music by Bach. But the sections of the book that are textual variations on Bach pieces quickly becoming a bit annoying and artificial (especially if you're a musician). However, they do provide a good alternative way of explaining math and number theory in a different way by describing it in terms of music, which helps a lot in understanding the whole story.
At the time, everybody that I tried to talk to about this book reacted "wow, way too difficult, do you really understand all that stuff?", but I've always felt that the only problem with it was its length. It's almost 800 pages, but Hofstadter does a very good job of explaining number theory in lots of different ways to make his point for non-mathematicians.
Or does that make me a geek? :-)
The "proof" of Gödel's theorem in Douglas Hofstadter's book is on page 271-272 in my edition (under the subheading: "G: a string which talks about itself in code"). The entire text up to that point explains number theory in layman's terms, and what Gödel's reasoning was in developing his theorem, so it's probably difficult to understand those pages if you haven't read what leads up to that point.
Part of what made GEB an interesting book for me to read was that it mixes computer and number theory with a description of how DNA encoding works, and with the music by Bach. But the sections of the book that are textual variations on Bach pieces quickly becoming a bit annoying and artificial (especially if you're a musician). However, they do provide a good alternative way of explaining math and number theory in a different way by describing it in terms of music, which helps a lot in understanding the whole story.
At the time, everybody that I tried to talk to about this book reacted "wow, way too difficult, do you really understand all that stuff?", but I've always felt that the only problem with it was its length. It's almost 800 pages, but Hofstadter does a very good job of explaining number theory in lots of different ways to make his point for non-mathematicians.
Or does that make me a geek? :-)
March 4, 2014 at 21:37 |
Nicole
Nicole
Is mathematics a discovery or an invention?