Mathematics: Invented or Discovered?

[Orbifold]

I agree with Spiritus Mundi, there’s way too much to catch up on…fortunately I also agree with everything Arnold Winkelried, panamajack, and Cabbage said, so that simplifies things.

*Originally posted by Nen *
panamajack, ask any mathematician whether or not mathematics is a language and I’m sure they will not assert that the comparison between mathematics and language is an analogy–mathematics is a language just as is an program or dialect.

Whoa there. I’m a mathematician, and I think that mathematics is more than a language. See Cabbage’s post. (Note that he’s a mathematician too.)

A tree does exist because it possesses physical properties. The value of pi does not exist without a formal system to define because 1) pi does not exhibit any physical properties, and 2) pi is solely defined by the construct.

Here’s an interesting point where we disagree. The formal systems of mathematics do not, at their heart, “define” numbers. The first of Peano’s axioms is “There exists a natural number 0”. That’s not defining “0”, in my mind: that’s defining a symbol to refer to a pre-existing thing.

Obviously, properties of an object to not appear at our whims, but one must differentiate between intrinsic properties and perceived properties. One can perceive two people standing next to each other; and therefore enumerate two people, yet the number two is intrinsic to the set of two people. Physically speaking, there is a flux in density. There may be, and most certainly are, properties which we have yet to observe, but these properties must be primary in nature, not secondary.

Cardinality is more than a “perceived property”. Take two protons and two anti-protons, toss them in a magnetic bottle, and wait for them to annihilate each other. There won’t be any protons or anti-protons left over, because the two sets have the same cardinality. If the two sets had different cardinalities, then some protons or some anti-protons would not get annihilated. So how can cardinality be just a “perceived property” of the two sets of particles? Certainly whether or not a particle remains after the experiment is a lot more than a “perceived property”!

**

*Originally posted by Math Geek *
But how can any language come into being without describing some pre-existing phenomenon?

Naturally, it can.**

Excuse me? If a language didn’t describe pre-existing phenomena, how the heck would anyone ever learn it? How would anyone ever share a language with anyone else?

Originally posted by Tymp
There is no function of my being that separates or groups encountered objects by number.

How many fingers do you have? Seriously, you seem to be suggesting that math is entirely invented because it’s something we think about rather than something that just happens, like chemical reactions with our taste buds or apples hitting the ground. The problem is, that’s not entirely true. Math happens all the time in the physical world without any concious thought occuring. In the example of protons and anti-protons above, subtraction is occuring. Whenever an enzyme docks with the appropriate molecule, a geometrical operation is performed, namely a comparison of two shapes.

These mathematical operations may not get interpreted until an observer comes along and thinks in the language of math – the operations may or may not be given meaning – but they’re still occuring. And the person describing the phenomenon is not inventing the phenomenon being described.

*Originally posted by Spiritus Mundi *
5) Physical objects have properties which we use mathematics to describe. They also have properties which we use English to describe. We discover the properties and invent the languages to express them.
…
8) The laws of physics are discovered. The language of physics is invented. (Come on – you knew that was coming.)

Your point #5 is exactly what I’m trying to say. I just claim that some of those properties are also part of “mathematics”…that math is more than the language. Again, see Cabbage’s post. And as for #8, just substitute “math” for “physics” and you have my position again. (Nen, this applies to your latest post as well.)

Cabbage, FriendRob: I forgot that there were two “Godel’s Theorems”, thanks for clearing that up. It’s interesting that both sides in this debate are using them.

Okay, back to work for me…

[panamajack]

*Originally posted by Nen *

The term “chair” is a very loose term, thus many thing can be included in its category. The term “circle” simply has a more precise definition. Were we to more rigorously define a set of objects with another word, say “Lazyboy,” the relationship that this-chair-is-a-recliner-produced-from-this-particular-manufacturer is unchangeable. Did we discover Lazyboys?

I’d actually thought about that before I read your post. The difference between even the strict definition of a Lazyboy and the mathematical properties is that the mathematical relationships could not have been otherwise from the start.

We can see that “this-chair-is-a-recliner-produced-from-this-particular-manufacturer” is unchangeable but that is only because we have actually defined the Lazyboy as “a reclining chair produced by the Lazyboy company of Monroe, Michigan”. In other words, the relationship was part of the definition.

I don’t see the circle’s relationship the same way. The circle is defined as the locus of points equidistant from a single point in a plane. Now where did the pi-ratio come from? It was not a stated property of the definition. Is it inherent in the plane, the point, or the property of distance?

In short, there is no other way that a circle could have been a circle. There is no other way to invent cardinality, either. If they could not be otherwise, we ought to say they are discovered.

Nen, you mentioned that the comparison of two trees is not a property of the trees. It certainly does not need to be. No one ever said mathematical properties are inherent to objects. They are, rather, about objects. It is abstract truths about objects that make up mathematics. The set of relationships is a property of the universe, so to speak.

[Spiritus_Mundi]

Has the debate shifted? Are those arguing for “discovery” now saying “partly discovered and partly invented”?

If so, I see little cause for dissent. Every invention is built from discovered components. Now, I am quite willing to quibble over whether any particular element is discovered (cardinality, pi, properties of planar graphs, etc.) but that seems a different set of questions, really.

[Tymp]

*Originally posted by Math Geek *
The formal systems of mathematics do not, at their heart, “define” numbers. The first of Peano’s axioms is “There exists a natural number 0”. That’s not defining “0”, in my mind: that’s defining a symbol to refer to a pre-existing thing.

That thing being nothing – nothingness, yes? How can nothing exist other than in relation to something? How can ‘0’ be more than a description of nothing in relation to something? I swear I’m not trying to be dense here, I’m just not sure what you’re after.

Excuse me? If a language didn’t describe pre-existing phenomena, how the heck would anyone ever learn it? How would anyone ever share a language with anyone else?

Music, friend. Music is a thing - a language - invented a long time ago. Still, people are discovering properties of music and the relationships between sounds that have always existed in the world of music but have never before been described and explored.

How many fingers do you have?

I still hold that quantity is not a physical property of anything. Finger, finger, finger, etc.

Seriously, you seem to be suggesting that math is entirely invented because it’s something we think about rather than something that just happens, like chemical reactions with our taste buds or apples hitting the ground.

That’s exactly what I am suggesting. In your example of protons and antiprotons, the physical characteristic of existence changes without observation or interpretation, but no subtraction is going on until an observer has a need to describe the event as such.

[Joe_Cool]

My position falls strongly in agreement with Math Geek and Arnold: The laws and relationships collectively described through the language we call “mathematics” have always existed, and will continue to exist after we’re long gone. The language is invented, but it was invented to describe relationships that were discovered.

pi may not have been named or calculated until some human decided to do it, but the relationship between a collection of points equally distant from a center has always existed. Planets have always followed their orbits, long before we even knew they had orbits. The Fibonnacci sequence and the golden ratio have appeared in spirals, trees, flowers, etc since those things existed, long before the sequence was named (yes I consider the sequence to be invented, but the properties intrinsic in it, namely the golden ratio, were discovered). If there’s only one way it could have been, then nobody invented it. Nobody “invented” the number of fingers, eyes, lungs, hearts, etc each person has. Nobody invented the number of trees in a forest. These things were discovered then given names. It’s true that the names didn’t exist until defined, but what is being described by the name has always been there.

*Originally posted by Tymp *
Arnold scratches in the sand and proclaims triumphantly, “Circle!” I scratch in the sand and declare the same. Arnold responds, “No, you idiot. That’s elliptical. A circle has a ratio blah, blah …” and I tune him out while drawing more pictures in the sand. Pi does not exist until Arnold wants to describe the difference between his sand scratching and mine.

Well, it doesn’t matter. pi is intrinsic to both the circle and the ellipse. The relationship has always been and always will be. The fact that we hadn’t observed the relationships doesn’t mean that they weren’t there.

I’m amazed at this discussion. It almost became a big “If a tree falls in the forest with nobody to hear it, does it make a sound” mental masturbation, but seems to have recovered nicely.

[Tymp]

*Originally posted by Joe_Cool *
The laws and relationships collectively described through the language we call “mathematics” have always existed, and will continue to exist after we’re long gone. The language is invented, but it was invented to describe relationships that were discovered.

Originally posted by the great and mighty Spiritus Mundi
5) Physical objects have properties which we use mathematics to describe. They also have properties which we use English to describe. We discover the properties and invent the languages to express them.

How is mathematics different from any other language invented to describe characteristics of the world?

*Originally posted by Joe_Cool *
yes I consider the sequence to be invented, but the properties intrinsic in it, namely the golden ratio, were discovered

If there’s only one way it could have been, then nobody invented it. Nobody “invented” the number of fingers, eyes, lungs, hearts, etc each person has. Nobody invented the number of trees in a forest. These things were discovered then given names. It’s true that the names didn’t exist until defined, but what is being described by the name has always been there.

How is quantity a natural physical property of an object? How is it, exactly, that the number of trees in a forest has a bearing on what a tree is? No matter how the trees are counted by an observer, the tree, tree, tree and tree are unaffected.

*Originally posted by Tymp *
Arnold scratches in the sand and proclaims triumphantly, “Circle!” I scratch in the sand and declare the same. Arnold responds, “No, you idiot. That’s elliptical. A circle has a ratio blah, blah …” and I tune him out while drawing more pictures in the sand. Pi does not exist until Arnold wants to describe the difference between his sand scratching and mine.

Well, it doesn’t matter. pi is intrinsic to both the circle and the ellipse. The relationship has always been and always will be. The fact that we hadn’t observed the relationships doesn’t mean that they weren’t there.

These circles intrigue me. I’ve heard all about them. I know all about their alleged properties, but never have I encountered one in nature. Perhaps you could suggest where I could observe a perfect circle. Perhaps you can also suggest a means by which I can verify that this example of a perfect circle meets precisely all the criteria that I have come to know to define a perfect, theoretical circle. As it stands, the only perfect circle I have ever encountered is the construct that resides in my mind and it is a most unnatural invention.

I’m amazed at this discussion.

As am I. It is most fortunate that all participants have maintained such focus and, more importantly, patience.

[panamajack]

Tymp, you seem to be saying that mathematical properties are invented because they are not observable physical properties of objects. So I’ll ask the question, why does something have to be attached to an object to be innate? The relationships are not changeable. They are discoverable properties of how things relate to one another. You could not expect them to be attached to a single object.

If you prefer, you could say they’re attached to just that system of objects, although that’s going only part of the way over. You’d have an inductive mathematics, at least (here’s two-trees, there’s two-monkeys, there’s some more two-trees, … until two-concept exists).

DISCLAIMER : the following is my own beliefs, possibly not germane to this debate, but I just felt I’d offer my perspective.

This inductive theory of math goes along with what you said about the imperfect circles everywhere. We can come to a concept of a physical circle by induction. To be honest, I’m somewhat in agreement with you (& Maeglin, and others) at this point. I believe Physics cannot be more than empirical.

The jump to mathematics is made because of our ability for abstract thought. We discover the mathematical property because we can think. Is abstract thought unnatural? I don’t think so.

Also, I’m not saying that mathematics is not prior to the existence of humans to think about it. That falls along the lines of Joe_Cool’s trees (how many trees were there, and how big where they?) falling in the forest, a sidetrack best left unfollowed. The relationships were there, whether or not anyone could comprehend them.

[Tymp]

The jump to mathematics is made because of our ability for abstract thought. We discover the mathematical property because we can think. Is abstract thought unnatural? I don’t think so. – panamajack

Mathematics results from abstract thought? I’ll accept that. Abstract thought is perfectly natural? I’ll accept that too. The fact that mathematics exists within the perfectly natural realm of abstract thought does not mean that mathematics exists in the world as a natural thing to be discovered in the absence of abstract thought that gave birth to it and upon which it is dependant.

Music was also born of very natural minds. Relationships between sounds that are defined and described by the concept of music exist regardless of any system used to define or predict them. This does not mean that music is a natural thing that exists in the world even in the absence of a mind that conceives of it, right? How is math different?

The relationships were there, whether or not anyone could comprehend them.

But the means of comprehension was invented.

[Orbifold]

*Originally posted by Tymp *
That thing being nothing - nothingness, yes? How can nothing exist other than in relation to something? How can ?0? be more than a description of nothing in relation to something? I swear I?m not trying to be dense here, I?m just not sure what you?re after.

Just that there was something pre-existing for the symbol “0” to refer to.

To be honest, I’m also not quite sure where you’re coming from at this point. Even if nothingness can only be defined in relation to something, I don’t see how that implies that we invented nothingness. Why can’t we discover a relation?

Music, friend. Music is a thing - a language - invented a long time ago. Still, people are discovering properties of music and the relationships between sounds that have always existed in the world of music but have never before been described and explored.

That’s an interesting point, but I’m just going to turn around and claim that some things being communicated from one person to another by music, emotions for example, were in a sense pre-existing, and then we’re just going to have to debate whether or not emotions were discovered or invented, so I don’t think this line of inquiry is going to get us very far

**I still hold that quantity is not a physical property of anything. Finger, finger, finger, etc.
[…]
That?s exactly what I am suggesting. In your example of protons and antiprotons, the physical characteristic of existence changes without observation or interpretation, but no subtraction is going on until an observer has a need to describe the event as such.
**

In the experiment I described the existence of the particles at the end of the experiments depends crucially on the cardinality of the sets of particles at the beginning. I just don’t see how a physical characteristic could depend on something which is just an invention of the observers.

As to your other point above regarding whether or not subtraction has occurred before an observer shows up, I explained my position on that in the paragraph following the part that you quoted.

This has been a wonderfully interesting debate (and it would be an oddly appropriate thread for my hundredth post), but I think we’re starting to spin our wheels a bit. Barring miraculous intervention, I don’t think either side is going to convince the other anytime soon…still, it’s fun to try!

[Cabbage]

Apparently we disagree on whether “quantity” is a physical property or not, so in addition to what I’ve said before, I thought I’d toss this out and see what kind of response it gets.

It’s well known the number of protons in the nucleus of an atom determines what element the atom is; additionally, the number of neutrons and electrons further determine the physical properties of the atom. This seems to me to be an explicit case of quantity determining physical properties, and I can see no place where it can be said that human invention played a part. How, then, can quantity itself not be considered a physical property?

[dixiechiq]

[QUOTE]
*Originally posted by panamajack *
**dixiechiq, for which side are you presenting evidence? The fact that the two geometries are incompatible means that there are situations where one or the other will simply not work in the ‘real’ AKA physical world, even if they both have applications to other parts of it.
**

lets assume you’re statement is correct. ie, two incompatible geometries that have differing application in the physical world.

that proves that neither geometry is discovered because neither of them is a full and correct description of the physical world. they are both just made up to try to describe it.

[dixiechiq]

for those that say mathematics is discovered…

i challenge you show me a circle or a line or a plane existing in nature. there are none, we invented all of those concepts. they do not exist. you can’t even make any of them!!

[Spiritus_Mundi]

I would really love to delve into these issues of whether cardinality and subtraction might be discovered elements, but I really would like to establish the basic proposition first. Do those arguing for discovery agree that the “strong discovery” position is untenable? Are we now arguing over whether specific elements might be discovered rather than whether mathematics as a whole is discovered?

[Tymp]

*Originally posted by Math Geek *
That’s an interesting point, but I’m just going to turn around and claim that some things being communicated from one person to another by music, emotions for example, were in a sense pre-existing, and then we’re just going to have to debate whether or not emotions were discovered or invented, so I don’t think this line of inquiry is going to get us very far

Actually, I think this gets us exactly where we need to be. Mathematics, like music or any other language, is invented to describe and comprehend extant aspects of the physical world that have been discovered. Further discoveries of aspects of the language can serve to support the effectiveness of the language but cannot serve to establish the language as a natural part of the world. Additionally, as new aspects of the natural world are discovered, the language is revised or expanded in order to encompass them, but the newly discovered things do not naturally possess elements of the language. Rather, they can be said to possess relations to the invented language that permit natural or rational revision of the language. Again, this does not establish that language itself is a natural thing that can be discovered.

Cabbage,

I happily and eagerly concede that the workings of the world are dependent upon systems.

Spiritus,

Yours is a fair assessment of where we have ended up, I think. It would be really nice if our friends in the discovery camp could offer some confirmation.

[Joe_Cool]

*Originally posted by dixiechiq *
**for those that say mathematics is discovered…

i challenge you show me a circle or a line or a plane existing in nature. there are none, we invented all of those concepts. they do not exist. you can’t even make any of them!! **

A circle is only a special case of an ellipsoid. I can show you ellipses existing all over the place: Planetary and stellar orbits.

I can show you a plane, as well: The orbital plane. Sure, not all planets in our solar system are in the same plane, but each planet has its own plane.

I can show you some things that are close enough to circular or spherical that without a micrometer, you wouldn’t know the difference: Look in the mirror. See those things in your head that allow you to see? Unless you’re grossly misshapen, your eyes (more specifically, the irises and pupils of your eyes) are very nearly circular.

Another example: Go outside at noon. Look up. You may want to shade your eyes. The sun, with the naked eye, appears very nearly circular. In fact, it’s elliptical as well. Plenty of places to find pi.

There are also plenty of places where you can find the golden ratio. And the fibonacci sequence. And e. And plenty of other numbers. None of those numbers were “invented”, they were discovered. Hell, had I invented pi, I’d have invented it to be an integer instead of some irrational, transcendental crap like it is now.

[Joe_Cool]

*Originally posted by Tymp *
[Spiritus Mundi’s] is a fair assessment of where we have ended up, I think. It would be really nice if our friends in the discovery camp could offer some confirmation.

Well, I’d be glad to restate what I said in the first place: Mathematics is a system invented by humans to describe and explain pre-existing relationships they saw in the world around them. Regardless of whether there was anybody to “hear the tree fall.”

So in a sense I guess we agree. The system of symbolic manipulation of quantities and relationships (what we call mathematics) was invented. But the quantities and relationships themselves were discovered, and were the reason for the invention of the system of manipulation. Is that good enough?

[Tymp]

Joe, I’m afraid every one of your examples of an ellipsoid is a failure. None of them can be said to meet the mathematical definition and it is not possible to verify the properties of one that you suspect to meet such a definition. Pi is an aspect of a constructed ideal, as has been stated repeatedly. This is what dixiechiq was getting at. However, that doesn’t really matter at this point.

[Nen]

I appear to be a bit behind in this conversation.

It is my understanding that the proponents of the discovery argument maintain that the formal system known as mathematics is an invention utilized to express things about discoveries, i.e., a language. It is also my understanding that the proponents of the discovery argument maintain that the formal system known as mathematics is more than a language. Maintaining both tenets simulataneously is contradictory.

Originally posted by Cabbage
In either case, however, it seems quite clear to me that the observable properties of the counting numbers dictated the choice of the axioms, not the other way around. Again, this indicates a discovery to me, not an invention.

What are the observable properties of counting numbers?

Originally posted by Maeglin
Our systems, of course, are limited by our perception. So for all we know, the fact that the laws of physics actually work could be mere coincidence. The laws of physics are useful because they work, to be sure. But it would be arrogant to presume that they are true just because they work, for it implies that our current perception of phenomena is so total that only our explanation could possibly hold under all conditions.

You are quite correct, sir. If Newton’s laws were true, how could they have been improved upon by Einstein? The language of physics, e.g., mathematical models, are approximations limited by our perceptions. That does not indicate that the laws of physics are invented, it simply implies that our expression of physics is not necessarily true. Essentially, the laws of physics are discovered, yet our invented abstraction is incomplete.

Originally posted by Math Geek
Whoa there. I’m a mathematician, and I think that mathematics is more than a language. See Cabbage’s post. (Note that he’s a mathematician too.)

Okay, so you’re a mathematician. I’m a physicist and philosopher. But how is mathematics more than a language?

Originally posted by Math Geek
The first of Peano’s axioms is “There exists a natural number 0”. That’s not defining “0”, in my mind: that’s defining a symbol to refer to a pre-existing thing.

I thought natural numbers were defined as the set numbers {0, 1, 2, 3, 4, 5, 6, 7, . . .}. Regardless, show me a zero. How does it exist?

I do not believe that zero exists. It is an abstraction–a construct. Languages can describe negative existentials, e.g., a unicorn.

I’ll also chime in with the others on this notion of pi. Someone, I beg of you, show me a circle. Don’t bother with a cross section of some kind of spheroid–that isn’t a circle. A circle is a perfection, and an abstraction, which does not exist. One can approximate a circle, yet circles do not exist. This wonderous relationship that is “unchangeable” exists only in theory.

Anyway, in order to proceed further in this discussion, I would appreciate it if someone would respond to the first part of my post.

[Orbifold]

*Originally posted by Spiritus Mundi *
**I would really love to delve into these issues of whether cardinality and subtraction might be discovered elements, but I really would like to establish the basic proposition first. Do those arguing for discovery agree that the “strong discovery” position is untenable? Are we now arguing over whether specific elements might be discovered rather than whether mathematics as a whole is discovered? **

Perhaps you could let us know to whom you’re addressing this question, specifically. I didn’t answer it the first time you asked because I thought that I had already stated my position pretty clearly…but if you want me to clarify myself, let me know.

*Originally posted by dixiechiq *
that proves that neither geometry is discovered because neither of them is a full and correct description of the physical world. they are both just made up to try to describe it.

No…I would say that implies that the universe exhibits at least two different sets of geometrical properties (depending on the circumstances, or on what aspect of the universe we’re observing), and that we’ve invented ways of describing them both. But we didn’t invent the properties that the universe exhibits in the first place.

Besides, gravity isn’t a “full and correct description of the physical world” either, but we still discovered gravity.

*Originally posted by Tymp *
Actually, I think this gets us exactly where we need to be. Mathematics, like music or any other language, is invented to describe and comprehend extant aspects of the physical world that have been discovered.

As I’ve stated to Nen once already, I feel that mathematics is more than the language used to express it. That’s why I’ve been arguing in this thread that the universe has mathematical properties which existed before our description of them.

Other than that I agree with what you’ve said above.

*Originally posted by Nen *
It is my understanding that the proponents of the discovery argument maintain that the formal system known as mathematics is an invention utilized to express things about discoveries, i.e., a language. It is also my understanding that the proponents of the discovery argument maintain that the formal system known as mathematics is more than a language. Maintaining both tenets simulataneously is contradictory.

To me, the statements “Math is just a language” and “Math is just a formal system” are pretty much the same. I don’t agree with either. I also don’t consider mathematics to be just the description; my notion of mathematics includes the phenomenon being described. Does that also answer your question, “how is mathematics more than a language”?

What are the observable properties of counting numbers?

Both Cabbage and I have posted examples to this effect. Although for the sake of accuracy I should say that those are examples of the observables properties of the phenomenon of cardinality, not the symbols we use to describe them. (It seems that distinction bears repeating.)

** I thought natural numbers were defined as the set numbers {0, 1, 2, 3, 4, 5, 6, 7, . . .}.**

There are other definitions of the natural numbers; for example, it’s possible to define number theory in terms of sets. Then “0” is defined to be the empty set, for example. But that kind of begs the question: what is the empty set? And if you look of the axioms of set theory, you’ll find that one of them is, “There exists the empty set”. Either way, there’s a reference to a pre-existing phenomenon.

**Regardless, show me a zero. How does it exist? **

I’ve been using the word “phenomenon” for a reason. No, “zero” doesn’t exist in the same way that rocks, bricks, or hammers exist. The laws of physics don’t exist in that way either. But they exist in another way, as does zero. You’ve said yourself that the laws of physics are discovered; I’m just saying that zero was discovered in the same way.

I’ll also chime in with the others on this notion of pi. Someone, I beg of you, show me a circle.

The circle thing is tricky, I admit. (Unfortunately, Joe_Cool, I think one of the consequences of general relativity is that planetary orbits aren’t perfect ellipses. Not the orbit of Mercury, at any rate. Somebody correct me if I’m wrong…) Euclidean geometry contains lots of things that we only see in the world as ideals, or as limits of other less perfect things. (A stick is almost straight, the path of a raindrop is much straighter, the edge of a crystal facet is straight almost to the limits of measurement, and so on.)

Which is why I’ve been talking about number theory. And other mathematical phenomena which do show up perfectly in nature, like fibonacci numbers (thanks for reminding me of that one, Joe).

(One hundred, one hundred posts, ha ha ha!)

[panamajack]

I apologize for not taking the time right now to fully respond to the first part of your post, Nen. My personal opinion is that it is not more than a language, it is just unique in that it is the only possible language.

I just need to catch up on something Tymp said on page 1.

Mathematics results from abstract thought? I’ll accept that. Abstract thought is perfectly natural? I’ll accept that too. The fact that mathematics exists within the perfectly natural realm of abstract thought does not mean that mathematics exists in the world as a natural thing to be discovered in the absence of abstract thought that gave birth to it and upon which it is dependant.

Well, I know you accept that, but I specifically stated the opposite view (that mathematics does not require humans to think of it). Mathematics is not a result of abstraction. Abstract thought is merely the means of discovery, like shape is discovered through visual or tactile perception.

My firm stand is that thought is just as valid a means of perception as sight or touch. And abstract relationships are a discovered part of the world. I think it is clear that most of those on the other side will not accept this. If I’m correct, the inventionists believe that anything thought must by definition be invented.

I doubt I can provide much evidence one way or the other, but consider that if all independent thinkers come up with an identical system (and not one varies), then it might be harder to say that the system was invented.

I’m afraid that at least I have reached the sticking point here, though I’m not averse to continuing.