Mathematical Platonism vs. Mathematical Formalism

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Lemminkainen
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Mathematical Platonism vs. Mathematical Formalism

Postby Lemminkainen » Tue Mar 02, 2010 5:09 am UTC

So, ladies, gentlemen, and people who don't firmly fall into predefined gender categories, after reading Neal Stephenson's Anathem, I feel compelled to discuss the issue titling the thread with some other intelligent people. So, do you take the Platonist perspective (in brief, that mathematical objects and relationships exist independently of human cognition and discovery) or the formalist perspective (that mathematical relationships are entirely the product of symbolic manipulations in axiom systems created by humans)?

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby LegoLogos » Tue Mar 02, 2010 6:01 am UTC

I personally find the formalist perspective the more sensible. Isn't it possible to derive a system which is consistent and complete, but which does not necessarily hold in the real world?

I am quite committed to the idea that abstract things are just that - abstract. They do not exist in the same way concrete things exist. The assumption that there is an eternal unchangeable object, to which the real world is merely an approximation, seems unnecessary.

Of course the mere fact that you have read a book on this subject ensures you are more qualified to comment than I am!

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby folkhero » Tue Mar 02, 2010 7:34 am UTC

I dunno if reading a book makes one qualified. At least in my case, I read Anathem and (much of) a book called "Platonism and Anti-Platonism in Mathematics," and I still feel like I have no idea one way or the other as to what I believe. It probably doesn't help that the later book first argued for platonism, then argued against it and then argued that the question was an incoherent one.
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby somebody already took it » Tue Mar 02, 2010 10:53 am UTC

Why should one preclude the other?

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby ianf » Tue Mar 02, 2010 1:11 pm UTC

Lemminkainen wrote:So, do you take the Platonist perspective (in brief, that mathematical objects and relationships exist independently of human cognition and discovery) or the formalist perspective (that mathematical relationships are entirely the product of symbolic manipulations in axiom systems created by humans)?


Interesting question. I've always taken the formalist perspective and didn't even think of the Platonist perspective. I've never thought of mathematical objects and relationships as things which exist (like LegoLogos, I think that abstract means abstract). And I therefore thought of mathematical entities as things which are invented rather than discovered.

I've also been interested in the relationship between mathematics and physics. I've always felt that mathematics does not explain nature, but rather describes it. I'm not sure I'm going to be able to explain this very well. I think that if you've got an equation which allows you to model a natural phenomenon and predict its future behaviour, there is a temptation to use phrases such as "planetary motion follows Kepler's Laws". However, a better statement would be that Kepler's Laws describe planetary motion. It's not that the Universe obeys some pre-existing mathematical rules, but humanity's invented mathematics do a good job of describing the Universe.

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby BlackSails » Tue Mar 02, 2010 1:13 pm UTC

Platonism all the way.

In the absence of man, the same things would still be true. As a blow against formalism, there are true things we cannot prove - that doesnt make them any less true

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Whimsical Eloquence » Tue Mar 02, 2010 5:37 pm UTC

BlackSails wrote:Platonism all the way.

In the absence of man, the same things would still be true. As a blow against formalism, there are true things we cannot prove - that doesnt make them any less true


In an absence of man there would be no Maths. Maths is nothing to do with Empirical Matters, like Science. Euclidean Mathematics, for instance, which was the very Mathematics championed by Plato has been shown to be incorrect over other forms of Geometry.

Mathematics is an art, an exercise of logic. It is reliant upon certain axioms. If these axioms are changed (there are different Mathematical fields with differing axioms) then the conclusions from them are destroyed.

Formalism for the win!
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby General_Norris » Tue Mar 02, 2010 7:33 pm UTC

"Maths" is an abstraction, it doesn't exist because it is not matter. My two cents.

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Lemminkainen » Tue Mar 02, 2010 7:57 pm UTC

BlackSails wrote: there are true things we cannot prove - that doesnt make them any less true


Care to give an example that's not assumed as an axiom in a mathematical system?

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby folkhero » Tue Mar 02, 2010 8:28 pm UTC

General_Norris wrote:"Maths" is an abstraction, it doesn't exist because it is not matter. My two cents.

Your definition of existence seems to suggest things like: planning, social norms, value, October 19th 1834 and the position of the president of the United States don't exist. I don't think many people would agree with such a limited definition of existence.

A question for the Platonists: Does the statement: "the Axiom of Choice is true" have an objective truth value? If it does, what is it? How would we go about finding out?
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Ingolifs » Tue Mar 02, 2010 8:50 pm UTC

Definitely formalism. Mathematics, like many other systems in society is composed of concepts and ideas. All are things we have developed over the milennia in order to interact with the natural world and get by with each other. Without humans, concepts such as art, love, truth, pi, morality, logic and so forth, would not exist.
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby General_Norris » Tue Mar 02, 2010 10:30 pm UTC

folkhero wrote:Your definition of existence seems to suggest things like: planning, social norms, value, October 19th 1834 and the position of the president of the United States don't exist. I don't think many people would agree with such a limited definition of existence.


Well, then what's the question? It's completely reliant on the definition. I stand by my opinion: Ideas don't exist by definition of "idea". Only matter can "exist". If ideas "exist" then there's no question.

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby polymer » Wed Mar 03, 2010 12:35 am UTC

I'd say I lean towards Platonism simply because mathematics is discovered, not invented. Although I'm generalizing when I say this. There are certainly aspects about mathematics that lend themselves to interpretation, and that are more or less difficult for a human to digest. Many tools in mathematics I feel are designed as a convenience for humans, but there are a consistent set of consequences that will arise no matter how you view the universe, and they reveal themselves depending on your axioms.

The Mandelbrot Fractal was not invented, it was discovered. The variety of ways you can look at it is infinite, in my opinion that isn't consistent with the idea of being invented.

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Raoul Duke » Wed Mar 03, 2010 3:27 am UTC

Ingolifs wrote:Definitely formalism. Mathematics, like many other systems in society is composed of concepts and ideas. All are things we have developed over the milennia in order to interact with the natural world and get by with each other. Without humans, concepts such as art, love, truth, pi, morality, logic and so forth, would not exist.


I think pi doesn't really fit well into that list, it's a little unsettling, I've always considered myself a formalist but this is irking me. I guess it comes down to whether circles exist in nature or are they solely abstract, our minds creating a perfect version of the near circles found in nature. Because of this close connection between the abstract and nature, is our abstract concept of the circle truly abstract? If circles aren't abstract then the ratio between their diameters and circumferences wouldn't be either and pi would exist (at least to a certain extent) in nature.

Another thing that got me thinking was the use of the word 'humans'. I don't know a huge amount about zoology but I think a couple if not most of those final concepts you named exist to an extent outside of humanity, in the animal kingdom. The human concepts of love and morality are just more advanced interpretations of the same concepts that kept us from killing each other back in the day, before the line between human and animal was as well defined as it is today, and they are still widespread through other species.

People often think of their minds are something unnatural, where the world of the abstract dwells, where concepts such as love, morality and logic float about, as products of our thought processes or innate. But if animals that we don't think of as having minds have some kind of concept of these notions how can they be completely abstract?

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby LegoLogos » Wed Mar 03, 2010 5:41 am UTC

Raoul Duke wrote:If circles aren't abstract then the ratio between their diameters and circumferences wouldn't be either and pi would exist (at least to a certain extent) in nature.


Why can't circles be abstract? Just because there are things in nature which can be approximated as being circles doesn't mean that the idea of a circle isn't abstract, does it? Not sure I have quite understood what you are saying here.

Raoul Duke wrote:People often think of their minds are something unnatural, where the world of the abstract dwells, where concepts such as love, morality and logic float about, as products of our thought processes or innate. But if animals that we don't think of as having minds have some kind of concept of these notions how can they be completely abstract?


Why is this question different to "if two people can understand the same abstract concept, how can it be completely abstract?" I was going to talk about this, but now I look closely I'm not sure I understand what point you are trying to make. Are you saying that because abstraction is the product of natural things (i.e. us) that they must be "real" or "exist" in the same way that natural things do?

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Le1bn1z » Wed Mar 03, 2010 7:14 am UTC

What about the Leibnizian/Newtonian compromise, that mathematical relationships are properties of existant being. Platonism, properly speaking, irreconcilably seperates the form of the thing from thing itself. For example, natural processes and formations can "participate" in the form, but never mimic it. The other side says that mathematics are an invention of reason. This last seems a contradiction, as reason is a faculty of understanding, rather than reasoning. Newton and Leibniz both concurred that mathematics were inherent in the essence of existence and of reason, that the proposition 2+2 = 4 is a condition of reason, a property of it like gravity to mass or extention to matter, rather than a creation of it or something seperated from it and beheld from a distance. Likewise, these fundamental mathematic relations are properties inherent in existence as much as extention or time, without which existence would be impossible.

This synthesis was cemented by Kant and as far as I am aware, there'd been no serious Platonic comeback in metaphysics, outside some obscure runs of Catholic or Orthodox theology. Did I miss something?
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Ingolifs » Wed Mar 03, 2010 9:47 am UTC

I think pi doesn't really fit well into that list, it's a little unsettling, I've always considered myself a formalist but this is irking me. I guess it comes down to whether circles exist in nature or are they solely abstract, our minds creating a perfect version of the near circles found in nature. Because of this close connection between the abstract and nature, is our abstract concept of the circle truly abstract? If circles aren't abstract then the ratio between their diameters and circumferences wouldn't be either and pi would exist (at least to a certain extent) in nature.


yeah, this is what I was getting at. Pi is a number, and like all numbers, is a concept. I remember struggling with this train of thought, while looking at how ubiquitous Pi is in physics and in patterns seen in the real world, that it surely must be something real and measurable - the ratio of a circle's circumference to its diameter. But then I reminded myself that I had never seen an actual perfect circle. I had merely seen objects that macroscopically resembled circles and approximated them to various degrees. Such a perfect circle can never exist in reality, simply because of the 'fuzziness' present at nano-scales as a result of quantum effects.

I hold the belief, that mathematics at its most basic and general, is simply the formalisation of processes of human thought. Even things as elementary as number and logic have their genesis as the cognitive tools humans developed as they attempted to survive in the african savannah. I do believe that these tools animals also posess to an extent - in that they can tell the difference between three objects and four objects, but cannot manipulate these ideas to deduce that a group of three plus a group of four make seven, say, or be able to count larger numbers without generalising to the concept of 'several' or 'many'.

Going back to the 'invent' vs 'discovery' issue in the debate, people seem to have a misconception that in order for formalism to hold, one would be able to 'invent', or in other words come up with all by themselves, complicated mathematical entities such as the mandelbrot set. I disagree with this, because I think that it is possible to discover something within any given system, without the system being some kind of immutable natural phenomenon. The best analogy I can think of is the discovery of a exploit or bug in a computer game. No one designed the computer or played it in a manner such as to 'invent' an exploit whose form was already fully thought up in the person's head. Rather, it is just simply discovered as a consequence of the game logic (most often then not, by someone accidentally stumbling across it or just by asking a simple 'what if?'). Another analogy would be the discovery of various persistent patterns found in the game of life. From pretty much any arbitrary system of rules, there can be consequences derived from the rules that aren't immediately apparent. This in no way indicates that there is something somehow eternal and immutable about said consequences.
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Le1bn1z » Wed Mar 03, 2010 4:31 pm UTC

I get the impression that some people misunderstand Platonism. I think some definitions are in order.

Three basic schools mentioned so far:

1.) Platonism. Mathematics is discovered by reason beyond nature, existing independantly of both nature and the mind.

2.) Formalism. Mathematics is created by the mind and imposed as a means of describing or understanding nature. Accepting Plato's statement that perfect mathematical exactitudes do not exist in nature, it states that their origin is not, as Plato believes, other worldly, but a human conceit.

3.) Aristoteleanism/Leibnizianism/Newtonianism. Mathematics is a characteristic or conditional element of both existant being and of reason, inseperable from either. Math is not created; it is a defining structural principle behind reason and existence, much like extension or time to being for existence (in Leibniz) or God (in Newton).

Declaring oneself to be Platonist because one "discovers" rather than "invents" math is a misnomer and misuse of the Platonist mathematics. Platonism is defined by a specific metaphysical understanding of the seperated relation of nature to ideas and forms. If you want to sound cool and Greek, you're an Aristotelean, believing that forms are inherent in the things themselves, a premise inhereted by Leibniz and Newton in the birth of the modern school of mathematical philosophy.

Much of modern science is premised on this third theory (until quantum theory and other whacked out advances of the 20th century, but that's another kettle of fish). I suspect most people fall into the third category today. In my life, I have only met five committed Platonists, and three of them ought to have been committed to a psych ward.
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Josephine » Thu Mar 04, 2010 5:30 am UTC

Does the definition of Platonism include all sentient civilizations? As I think I can be sure that alien civilizations would have a concept of math just as much as we do. But I think that that statement alone lends its hand to Platonism.
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby BlackSails » Thu Mar 04, 2010 5:45 am UTC

I really like Erdos' idea of "THE BOOK" where God keeps all the best proofs.

Now the expression of math, thats totally invented.

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Le1bn1z » Thu Mar 04, 2010 6:28 am UTC

nbonaparte wrote:Does the definition of Platonism include all sentient civilizations? As I think I can be sure that alien civilizations would have a concept of math just as much as we do. But I think that that statement alone lends its hand to Platonism.


Yes, the Platonists believe precisely that.

So do the Aristoteleans and Newtonians.
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Josephine » Thu Mar 04, 2010 10:43 am UTC

Le1bn1z wrote:
nbonaparte wrote:Does the definition of Platonism include all sentient civilizations? As I think I can be sure that alien civilizations would have a concept of math just as much as we do. But I think that that statement alone lends its hand to Platonism.


Yes, the Platonists believe precisely that.

So do the Aristoteleans and Newtonians.

Am I missing something here?
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Le1bn1z » Fri Mar 05, 2010 5:37 am UTC

nbonaparte wrote:
Le1bn1z wrote:
nbonaparte wrote:Does the definition of Platonism include all sentient civilizations? As I think I can be sure that alien civilizations would have a concept of math just as much as we do. But I think that that statement alone lends its hand to Platonism.


Yes, the Platonists believe precisely that.

So do the Aristotleans and Newtonians.

Am I missing something here?


Nope. Quite the opposite. That is definately a core aspect of the Platonic definition of reason: rational concepts, or forms, exist independently of a mind to think it. Minds are entities which can concieve of these independent forms. Therefore, these forms would never, ever differ mind to mind, no matter how different the minds, only thier relation to one-another or the perfection of the understanding of these forms. Plato was one of the earlier thinkers to equate evil with ignorance.

However, Platonism is not unique in this belief. There exists a broad middle ground between formalism and Platonism, where people like Newton, Leibniz and Aristotle all fit in, and they also believe in universal reason, no matter how different the minds.
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Qaanol » Fri Mar 05, 2010 6:16 am UTC

The concept of unity, or ‘one-ness’. That’s about as simple as math gets. What are the important properties of what we consider to be ‘one’, and do those properties exist without someone to think of them?

We consider the universe to be one thing, hence its name. If the universe contained no sentience, that is, if the universe did not ‘know itself’, would it cease to be one thing?

Moreover, would the universe would still exist if all intelligence were absent?

Please answer that last question. It may seem off-topic, but I have a very relevant reason for asking it.
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Le1bn1z » Fri Mar 05, 2010 6:34 am UTC

Qaanol wrote:The concept of unity, or ‘one-ness’. That’s about as simple as math gets. What are the important properties of what we consider to be ‘one’, and do those properties exist without someone to think of them?

We consider the universe to be one thing, hence its name. If the universe contained no sentience, that is, if the universe did not ‘know itself’, would it cease to be one thing?

Moreover, would the universe would still exist if all intelligence were absent?

Please answer that last question. It may seem off-topic, but I have a very relevant reason for asking it.


And that reason is........?

The physical nature of the thing is in the thing itself, not in the phenomenal nature of thing as precieved as something beyond itself. So, in brief, yes, it would still exist if there were no intellegent life. I don't know if it could be considered to be "one thing," as the unity of the universe is conceptual. It is a compilation of forces, particles and systems interacting with one another.

Now, how is this relevant to anything?
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Qaanol » Fri Mar 05, 2010 8:06 am UTC

You make a statement of the form “I think X”.

When X is “Y would be true if I did not think” then the whole statement is of the form “I think Y would be true if I did not think”, which is, by the principles of logic, equivalent to “I think (Y is true or I think).”

This means if you think, then it is true, and if you do not think, then it is false. It is therefore equivalent to “I think”. Thus X does not contribute anything to the statement. As a result, both the statements, ‘I think the universe would exist if I did not think’ and ‘I think the universe would not exist if I did not think’ are equivalent to each other and to the statement ‘I think’.

Therefore, in order to proceed, we need to know if logical statements have can have definite truth values in the absolute sense, without an implicit “I think” at the beginning that ties the result to a sentience. If they can, then logic exists independent of anyone thinking of it. Mathematics stems directly from logic, indeed logic may be considered a part of mathematics, so in this case mathematics, or at least parts of it, exists in the abstract. Also, in this case it is possible to have a definite answer to whether the universe would exist without sentience. Since logic and math would have to exist, it's hardly a stretch to say other things could too.

On the other hand, if no logical statement can have a definite truth value in the absolute sense, then logic only exists as a sentience thinks it. Then there is no way to know if the universe would exist without sentience. More to the point, if logical statements cannot be evaluated in the absolute sense independent of any sentience, then the statement “Logical statements cannot be evaluated in the absolute sense independent of any sentience” can only be evaluated by a sentience. But in the case where no logical statement has a definite truth value, that statement has been evaluated, and is true. Therefore this case can only occur if there is sentience.

Got that? Only if there is sentience is it possible for logic to rely on sentience. In that case, removing all sentience would necessarily render all logical statements as having no definite truth value. That includes the statement “There exists a logical statements which has a definite truth value”, which in this case has a definite truth value of false, but also must have no definite truth value. Therefore the statement “Logic statements only have definite truth values if there is sentience” would lead to a contradiction if it were true, so it cannot be true.

Ergo, logical statements can have definite truth values even without sentience. As I mentioned, logic is part of math, so at least some math exists even without sentience.

This entire argument was developed within the logic that we as sentient beings agree upon. Therefore either our logic is self-contradictory, or it exists independent of us. As a direct result, any mathematical system which is not self-contradictory must at least contain portions that are abstract and independent of sentience. We call a non-self-contradictory system ‘consistent’.

Gödel proved that a consistent system of sufficient complexity (such as our mathematics) must be incomplete. That means there are both true statements we cannot prove, and false statements we cannot disprove. Such statements are obviously independent of sentience, since sentience is insufficient to evaluate them, even though they have definite truth values to evaluate.

In conclusion, if our mathematics is consistent, then at least some parts, including basic logic, are abstract and exist independently of whether they have been thought of by anyone.
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Greyarcher » Fri Mar 05, 2010 10:15 am UTC

Qaanol wrote:This means if you think, then it is true, and if you do not think, then it is false. It is therefore equivalent to “I think”. Thus X does not contribute anything to the statement. As a result, both the statements, ‘I think the universe would exist if I did not think’ and ‘I think the universe would not exist if I did not think’ are equivalent to each other and to the statement ‘I think’.
If you translated those two sentences--with clearly distinct and contrary meanings--into formal logic and concluded that they are equivalent, I think this indicates a poor translation rather than actual equivalence of sentential content.

Qaanol wrote:Therefore, in order to proceed, we need to know if logical statements have can have definite truth values in the absolute sense, without an implicit “I think” at the beginning that ties the result to a sentience. If they can, then logic exists independent of anyone thinking of it.
Do you not see how insensible it is to see if we can remove the implicit "I think" from logical statements by using logical statements? Either the entire demonstration is undermined by the fact that there is always an implicit "I think" within the statements of the methodology, or there is nothing to prove in the first place because we are already capable of removing the implicit "I think" from the statements of the methodology, thus removing any need for the demonstration!

Qaanol wrote:Only if there is sentience is it possible for logic to rely on sentience. In that case, removing all sentience would necessarily render all logical statements as having no definite truth value. That includes the statement “There exists a logical statements which has a definite truth value”, which in this case has a definite truth value of false, but also must have no definite truth value. Therefore the statement “Logic statements only have definite truth values if there is sentience” would lead to a contradiction if it were true, so it cannot be true.
Except when you say it is false and without truth value, you're conflating the sentence as it would exist without sentience (no truth value) and the sentence as observed by us (false), a sentience. To put it another way, you tried to observe the truth value of the sentence while claiming the sentence's truth value is unobserved. And that's nonsense.

Qaanol wrote:This entire argument was developed within the logic that we as sentient beings agree upon. Therefore either our logic is self-contradictory, or it exists independent of us.
Or your methodology was bad, which is what it looks like to me.
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Le1bn1z » Fri Mar 05, 2010 3:52 pm UTC

Qaanol,

You're living in the wrong millenium. You would have been very happy in Ancient Greece, with the preSocratics, but I'm afraid you're on a track that's a little out of date. Its last gasp was in the middle ages. Or today in Iran, where medieval philosophy is alive and well.

Unfortunately, Pythagoras, Newton and Kant eached demolished any remaining dignity of this argument in their times.

They noticed:

1.) There is a world. We observe the world, but do not, thereby, control it. The world is independent of us.

2.) In the world, things exist and have properties.

3.) These properties are formulaic, like the mathematical description of a conch shell.

4.) These properties may be described through mathematics or logic.

The inability of our minds to describe a phenomenon in nature in no way negates that things' existence. Otherwise, the world would cease to exist at the point any one of us died, or developed brain damage.

Happily, there are billions of people, some of differing mental states, which allows us to accumulate verifiable information on the role of perception. As we are independent intellegences, and the world remains a neutral truth despite differences in perception, and preserveres despite the loss of a percieving being, we can rest assured that the universe of facts is not a stray strand of human logic.

Unless you're a brain floating in a vat. But that raises more questions than it answers.
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Qaanol » Mon Mar 08, 2010 9:14 am UTC

If the logic we use is valid regardless of the presence of intelligence, then our logic exists in the abstract. In this case I claim mathematics, being directly founded on logic, also exists in the abstract.

On the other hand, if the logic we use is only valid in the presence of intelligence, it would seem that our logic does not exist in the abstract. However, that conclusion would in fact be an application of modus ponens, which is a part of the logic we use.

I asked about the universe because “The universe exists” is a statement that, in our logic, has a truth value. If that statement is also to have a truth value without the presence of intelligence, then statements have truth values without the presence of intelligence.

Or, that would be the case if our logic applied. If our logic does not apply, that conclusion cannot be drawn, so “The universe exists” could have a truth value, and at the same time it might be that no statement has a truth value.

If we accept that “The universe exists” would be true, and we also accept modus ponens, so that “P implies Q” being true and “P” being true together entail that “Q” is true, then statements would have truth values.

If statements did not have truth values, then questions like “Is mathematics abstract?” and “Does the universe exist?” would be meaningless. Or at least that would be the case given modus ponens.

Do you see the line I’m walking here? If logic did not exist, then we, working within logic, could not draw any inference whatsoever. The statement “Mathematics exists” would be neither true nor false. Which means, within our logic, that mathematics would not exist as we know it, because if it did, that statement would have a truth value.

Did you catch that? If logic did not exist without us, then we, working within logic, would know simultaneously that both “Mathematics exists” is false, and that it is neither true nor false. That is to say, within our logic, there would be a contradiction, which means we could prove anything. Our mathematics would be worthless.

To rephrase, still within the logic we use, if our logic only exists when we are around, then our logic is not consistent. Therefore, if our logic is consistent, then it must exist in the abstract.

The moment we accept that statements would have truth values even without sentience to appreciate them, and that implications actually entail their consequences when their antecedents are true, then we have logic as we know it, in the abstract.

Of course, I am as always working within our logic, which means if our logic is inconsistent, the opposite of everything I have shown could also be shown. However, if our logic is consistent, then as shown it also exists in the abstract. Floating in a vat I cannot distinguish between an inconsistent logic in which we have not yet found a contradiction, and a consistent logic. Thus, I cannot say for sure whether our logic exists in the abstract. But I can say that if our logic is valid, then it is also universal and abstract.

The above holds if our logic is consistent, and the same is true about this sentence.

I posit that mathematics, or at least major parts of it, follows directly from logic.
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Zamfir » Mon Mar 08, 2010 10:57 am UTC

Qaanol, what about "The Merchant of Venice"? Would it exist in absense of minds, or even the universe? I would say that mathematics exist in the same abstracted sense that a play exists seperately from paper or stage, and that if mathematics 'exists' outside of the universe, then "The merchant of Venice" does too.

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Thrind » Tue Mar 09, 2010 1:29 am UTC

It seems there is a mixup between possibly different meanings of 'exist'. Physical things do not disappear if no humans are there to observe them. For abstract things like social norms, October 19th 1834 and pi however, this point is a bit harder to make. I don't see how social norms can exist in any unplatonic way without a society in the first place. And once you have concluded that, there's no substantial difference between a social norm and the Mandelbrot set. So while I'm thinking about them, these things do exist in some way, but it's different from the way the keyboard I'm typing on exists.

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Qaanol » Tue Mar 09, 2010 2:23 am UTC

In the case where our logic is inconsistent, it can be shown that both The Merchant of Venice does and does not exist now, and would and would not exist without sentience. So that is not an interesting case.

In the case where our logic is consistent, then the statement “The Merchant of Venice exists” probably has a definite truth value now. We generally agree that it is true right now, inasmuch as anything incorporeal exists. It could of course be paradoxical or not well-defined right now, although I rather doubt that. Similarly, that same statement about the existence of The Merchant of Venice would also likely have a definite truth value without sentience. I make no claims as to what that truth value would be. And just as the truth value of statements like “There is a bagel on my plate” can change over time, it is possible the current truth-value of the existence of The Merchant of Venice may change.

The important thing for showing that consistent logic implies abstract logic is the self-referentiality of statements about logic. That allowed a contradiction to arise one case but not in another. If you can show a contradiction to the existence or lack thereof for The Merchant of Venice in various situations (or reason to one or the other from first principles) then we could say that its existence follows from having a consistent logic. Personally, I rather doubt that The Merchant of Venice is a deterministic consequences of the basic axioms of logic, so its existence probably can’t be deduced by my methods. There may be other methods though.

Thrind: my point is that the very statement “the keyboard exists” is abstract in the same way as the Mandelbrot set. If you accept that “the keyboard exists” has a definite truth value even without sentience, you’re already conceding the independent existence of something abstract. Further, I’ve shown that if our logic is consistent, then it exists as such even without sentience. I claim that significant portions of mathematics follow directly from logic.
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Thrind » Tue Mar 09, 2010 7:40 am UTC

Qaanol wrote:If you accept that “the keyboard exists” has a definite truth value even without sentience, you’re already conceding the independent existence of something abstract.


I wouldn't accept that. In fact, you do not have to go that far in your argument, as the statement "the keyboard exists" is already something abstract, there's no need to talk about the truth value. In the magnificient absence of sentience, that statement would exist as much as the much loved Mandelbrot set.

I happily concede that much of mathematics follows from logic. That's about all that formal logic is good for, in fact. It was designed to build mathematics on, applying things like the existence operator in statements that are not axioms or follow from axioms is usually not very rewarding.

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby folkhero » Tue Mar 09, 2010 8:37 am UTC

When I think about these things in detail, I generally start agreeing with Quine's rejection of the synthetic analytic distinction. The fact that 1+3=2+2=4 is really an observation about how the universe works. The axioms used to prove such statements are useful only insofar as useful statements about how the world really is, or at least our sense experience of the world. Just as any scientific paradigm's usefulness is judged based on how well it gives consistent predictions and explanations of our experiences. If we find a domain where 2+8=3+4=7, provides a more accurate and coherent account of our sense data than 2+8=10 then we can and should dispose of the axioms of arithmetic when dealing with events and objects in this domain. This outlook puts mathematics in the same category as science, to be evaluated in the same way. Mathematics is so foundational, that the evidence would have to be overwhelming for us to change our mathematics at the levels of the axioms, where the important changes of mathematics would take place, but they aren't outside of scrutiny entirely.

Is this making sense, or do I need to get some sleep?
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Zamfir » Tue Mar 09, 2010 8:54 am UTC

folkhero wrote: Just as any scientific paradigm's usefulness is judged based on how well it gives consistent predictions and explanations of our experiences. If we find a domain where 2+8=3+4=7, provides a more accurate and coherent account of our sense data than 2+8=10 then we can and should dispose of the axioms of arithmetic when dealing with events and objects in this domain.

I'd say the important 'sciency' thing in mathematics is finding out which structures are worthy of studying and developing. There is no way how 2+8 equals 7, for our current definitions of 2, 8, 7 and +. But is possible to define objects that behave more or less like natural numbers and addition, but with differences such that the equivalent of "2+8=7" would be true with those different objects.

It's a scientific, empirical 'fact' that our current system is more useful to understand a wide basis of phenomena. But once you define the objects, their properties with respect to each other follow from those definitions, and it makes no sense to say that they could have different properties.

In this view, mathematical constructs are designed, just as a Toyota Camry is designed or a play is written. After you designed a car, written a play or came with an algebraic structure, there might be properties of your design that you didn't consciously put in. It is tempting to say that those properties "were always there", and that you are then discovering eternal mathematical truths if you find them.

Ifyou design a Toyota, and crash tests show that this particular design can't withstand a 50 km/h crash, you have also discovered an 'eternal truth' about the universe, and all other indentical car designs in all corners of the universe will have that property too.

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Le1bn1z » Tue Mar 09, 2010 1:27 pm UTC

Qaanol,

Don't forget that logic is merely a descriptive faculty; not a creative one. The laws of logic are necessarily true wherever there is intelligence, and can effectively describe a world which is replete with formal, regular structure which can be described by mathematics.

This does not, however, mean that 1.) the universe's existence, or the stability of its properties which can be described mathematically, are dependent on an intellegence's ability to interpret them propery and 2.) Simply because mathematics holds true in our minds and in the existant universe does not mean that math or logic exists on some ethereal plane.

It is absurd to think that math exists beyond mind or matter, as mathematics is, properly speaking, a defining property of both mind and matter. Seperating math from mind or matter is like sperating exension from matter and space: patently absurd.

I think you need to reconsider what you mean by "exists in the abstract." Since you use the techno jargon, I won't hold back. Your thesis implies that logic has a seperate, formal, unified and substantive existence in which being and intelligence may participate, but which is still beyond both intelligence and being.

To which I ask this. In what way can math exist without being either thought, or being contained in an existant substance. Or, to put it bluntly, try to concieve of mathematics without thinking formulae or drawing a picture.
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby Greyarcher » Tue Mar 09, 2010 8:31 pm UTC

Qaanol wrote:If the logic we use is valid regardless of the presence of intelligence, then our logic exists in the abstract. In this case I claim mathematics, being directly founded on logic, also exists in the abstract.

On the other hand, if the logic we use is only valid in the presence of intelligence, it would seem that our logic does not exist in the abstract. However, that conclusion would in fact be an application of modus ponens, which is a part of the logic we use
When people use the term, "abstract", I suspect they are not speaking of some special realm of existence for our thoughts, like Platonic Forms. I think the common usage is that if something "exists in the abstract" then that thing does not have material substance, but is merely thought. So, of the first statement, it would be more unambiguous to say, "If the logic we use is valid--hence, exists--regardless of the presence of our intellect, then our logic exists in some sort of Platonic realm of Forms".

It is more natural to say, of the second statement, that "If the logic we use is only valid (i.e. meaningful, extant) in the presence of intelligence, then our logic exists only in the abstract". In other words, it does not have material substance, but only exists in thought.
--It is unnecessary to posit some special realm of existence; logic is simply our brain's way of structuring its understanding of the world.

Qaanol wrote:I asked about the universe because “The universe exists” is a statement that, in our logic, has a truth value. If that statement is also to have a truth value without the presence of intelligence, then statements have truth values without the presence of intelligence.

Or, that would be the case if our logic applied. If our logic does not apply, that conclusion cannot be drawn, so “The universe exists” could have a truth value, and at the same time it might be that no statement has a truth value.
We can say "The universe exists", and because we exist we can say it has a truth value--the statement and the truth claim have meaning because it is made within the discourse of sentients. When sentients exist, they can make statements, apply logic, and make truth claims.

It is hopeless to talk of the truth value of a statement in a hypothetical Universe Without Sentients--a UWS has no truth claims or statements. Logic and truth values never come up in a UWS. We could say that 2+2=4 will be true even in a UWS, but that's like saying that music will still sound good in an absolute vacuum, or that unicorns will somehow "exist" in a UWS with no one to conceive them.

Qaanol wrote:If we accept that “The universe exists” would be true, and we also accept modus ponens, so that “P implies Q” being true and “P” being true together entail that “Q” is true, then statements would have truth values.

If statements did not have truth values, then questions like “Is mathematics abstract?” and “Does the universe exist?” would be meaningless. Or at least that would be the case given modus ponens.
Rather than meaningless or true, those statements and logic would not even exist in a universe without sentients, and there would be no one to call them meaningless or true. Thinking requires thinkers--statements, logic, truth claims, and truth values require thinkers.

Qaanol wrote:Do you see the line I’m walking here? If logic did not exist, then we, working within logic, could not draw any inference whatsoever. The statement “Mathematics exists” would be neither true nor false. Which means, within our logic, that mathematics would not exist as we know it, because if it did, that statement would have a truth value.
You are either conflating a UWS and our world-with-sentients again, or introducing confusion because of a Platonic notion about what it means for logic to "exist". Especially apparent in the second sentence in the final paragraph. "If logic did not exist, then we, working within logic"--obviously, by the structure of the statement itself, logic exists. Not in a special Platonic form, but as a way in which we sentients think and structure our thought. Mathematics is the same.

Qaanol wrote:Did you catch that? If logic did not exist without us, then we, working within logic, would know simultaneously that both “Mathematics exists” is false, and that it is neither true nor false. That is to say, within our logic, there would be a contradiction, which means we could prove anything. Our mathematics would be worthless.
And this confused conclusion is of the exact same nature as your previous post I replied to.

Essentially what you're doing is just different forms of this:
1. "Hey let's talk about a world without sentients."
2. "That world wouldn't have truth values because there'd be no sentients."
3. "Oh, but 'the world is flat' would still be false! Contradiction!"

Except instead of "The world is flat" you used--in the first post--"There exists a logical statements which has a definite truth value" and--in the second post--"Mathematics exists".

It's still the same error. The meaning to the hypothetical "world without sentients" is that, in that world, there is no one to make statements and truth claims, and therefore no basis for truth values either. Therefore, when you say, "'The sky is blue' is still true!" or "'Mathematics exists' would be false!" you are fundamentally misunderstanding what the hypothetical is illustrating.
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Re: Mathematical Platonism vs. Mathematical Formalism

Postby somebody already took it » Tue Mar 09, 2010 9:44 pm UTC

Le1bn1z wrote:To which I ask this. In what way can math exist without being either thought, or being contained in an existant substance.

Wikipedia:Gödel's incompleteness theorems wrote:The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (essentially, a computer program) is capable of proving all facts about the natural numbers. For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.

To relate this to your question, there are theorems which exist (in the sense that they are true) but cannot be arrived at by any algorithm (and unless human thought/existent substance is capable of deciding things algorithms cannot, which would violate the Church-Turing thesis, such theorems cannot be thought or contained in an existent substance).

Le1bn1z wrote:Don't forget that logic is merely a descriptive faculty; not a creative one. The laws of logic are necessarily true wherever there is intelligence, and can effectively describe a world which is replete with formal, regular structure which can be described by mathematics.

Is it possible to prove consistency for the laws of logic?

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby BlackSails » Wed Mar 10, 2010 12:03 am UTC

somebody already took it wrote:Is it possible to prove consistency for the laws of logic?


Sure, http://en.wikipedia.org/wiki/G%C3%B6del ... ss_theorem

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Re: Mathematical Platonism vs. Mathematical Formalism

Postby somebody already took it » Wed Mar 10, 2010 2:33 am UTC

BlackSails wrote:
somebody already took it wrote:Is it possible to prove consistency for the laws of logic?


Sure, http://en.wikipedia.org/wiki/G%C3%B6del ... ss_theorem

Is there a specific section of that article you had in mind?
Also, what are the laws of logic?


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