## Synthetic a priori, mathematics, and not-self

- Pannapetar
**Posts:**327**Joined:**Wed Jul 29, 2009 6:05 am**Location:**Chiang Mai, Thailand-
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### Synthetic a priori, mathematics, and not-self

The not-self (anatta) doctrine of Buddhism takes two forms: a special form and a generalised form. In the the special form, not-self means not belonging to me, not mine, not I, and it concerns false identifications of self with, for example, the five aggregates. In the generalised form, it states that all phenomena are not-self, meaning all phenomena are empty of inherent existence, that there is no elusive entity behind phenomena.

Now, there is a curious class of statements known to philosophers as a priori statements. Kant (who invented the term) has defined this as the class of statements that do not rely upon empirical verification. In other words, these are "logical" propositions that cannot be contradicted without violating logic. "All triangles have three sides" is an example of an a priori statement. More precisely, this is an example of an analytic a priori statement. Unfortunately, most analytic statements are quite boring. The other type is called synthetic a priori statement and these tend to be more interesting. For example, "11 + 9 = 20" is such a statement (though it could be seen as a border case because it follows directly from the definition of natural numbers). Anyway, take for example: C/2r= pi. The circumference of a circle divided by its diameter (radius *2) equals pi. There you have an example of a non-trivial synthetic a priori proposition.

You might begin to wonder why I am mentioning this. Well, the class of synthetic a priori statements appears to describe atta/atman, or perhaps better: the essence of things. For example, C/2r=pi describes the essence of the circle. For all we know, it is eternal, universal, and non-changing (within the confines of Euclidean geometry). Furthermore, pi plays an important role in various mathematical and natural laws ranging from physics to statistics. Conclusion: there are certain atman-like properties that we can know about certain phenomena. This brings up a number of interesting questions.

1. Are phenomena themselves manifestations of atta/atman as far as these laws are concerned?

2. Are there classes of mathematical laws that are subject to some kind of impermanence?

3. Are there physical laws that are subject to some kind of impermanence?

4. Does our mind/thoughts somehow experience atta/atman to the extent to which we can understand these laws?

5. What part of ourselves understands these laws?

6. An argument favoured by theologians (rejected by Buddhists): is it our atman/brahman nature that discovers and understands these laws?

...

Cheers, Thomas

Now, there is a curious class of statements known to philosophers as a priori statements. Kant (who invented the term) has defined this as the class of statements that do not rely upon empirical verification. In other words, these are "logical" propositions that cannot be contradicted without violating logic. "All triangles have three sides" is an example of an a priori statement. More precisely, this is an example of an analytic a priori statement. Unfortunately, most analytic statements are quite boring. The other type is called synthetic a priori statement and these tend to be more interesting. For example, "11 + 9 = 20" is such a statement (though it could be seen as a border case because it follows directly from the definition of natural numbers). Anyway, take for example: C/2r= pi. The circumference of a circle divided by its diameter (radius *2) equals pi. There you have an example of a non-trivial synthetic a priori proposition.

You might begin to wonder why I am mentioning this. Well, the class of synthetic a priori statements appears to describe atta/atman, or perhaps better: the essence of things. For example, C/2r=pi describes the essence of the circle. For all we know, it is eternal, universal, and non-changing (within the confines of Euclidean geometry). Furthermore, pi plays an important role in various mathematical and natural laws ranging from physics to statistics. Conclusion: there are certain atman-like properties that we can know about certain phenomena. This brings up a number of interesting questions.

1. Are phenomena themselves manifestations of atta/atman as far as these laws are concerned?

2. Are there classes of mathematical laws that are subject to some kind of impermanence?

3. Are there physical laws that are subject to some kind of impermanence?

4. Does our mind/thoughts somehow experience atta/atman to the extent to which we can understand these laws?

5. What part of ourselves understands these laws?

6. An argument favoured by theologians (rejected by Buddhists): is it our atman/brahman nature that discovers and understands these laws?

...

Cheers, Thomas

- Prasadachitta
**Posts:**974**Joined:**Sat Jan 10, 2009 6:52 am**Location:**San Francisco (The Mission) Ca USA-
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### Re: Synthetic a priori, mathematics, and not-self

Hi Thomas,Pannapetar wrote: For example, C/2r=pi describes the essence of the circle. For all we know, it is eternal, universal, and non-changing (within the confines of Euclidean geometry). Furthermore, pi plays an important role in various mathematical and natural laws ranging from physics to statistics. Conclusion: there are certain atman-like properties that we can know about certain phenomena. This brings up a number of interesting questions.

1. Are phenomena themselves manifestations of atta/atman as far as these laws are concerned?

2. Are there classes of mathematical laws that are subject to some kind of impermanence?

3. Are there physical laws that are subject to some kind of impermanence?

4. Does our mind/thoughts somehow experience atta/atman to the extent to which we can understand these laws?

5. What part of ourselves understands these laws?

6. An argument favoured by theologians (rejected by Buddhists): is it our atman/brahman nature that discovers and understands these laws?

...

Cheers, Thomas

I am currently putting off my mediation to play with your musings. It sounds like fun. I am not classically trained in logic, math, or philosophy so I have little to no qualifications by which my answers need be taken seriously. It just strikes me as a fun thing to ponder.

1) I think numeric laws are simply a form of communication which correspond to phenomena. Without a circle there would be no pie. Sort of like a recipe for cupcakes. Somebody discovered a thing and called it a cupcake then they studied the qualities and procedures that made up a cupcake. The recipe can reproduce a cupcake if it is understood properly. No great mystery. When an artist draws a circle they do not need to understand how to interpret "C/2r=pi" but they have in some way created a spatial relationship which later can be described in that way.

2)No circle= no "C/2r=pi"

3)No perception of a particular physical event= no law to describe it

4)As Far as I can tell I have never "experienced atta/atman". For me its like a word with nothing to correlate with but confusion.

5)When we create a model for an event or particular type of relationship and we remember that model we are said to understand it. Whatever parts are necessarily involved in this process are thus.

6) What we call discovery has its own specific conditions. Some conditions come together and there is understanding. Where is the confusion?

This has been a blast.

Thanks.

Time to sit....

metta

Gabe

"Beautifully taught is the Lord's Dhamma, immediately apparent, timeless, of the nature of a personal invitation, progressive, to be attained by the wise, each for himself." Anguttara Nikaya V.332

### Re: Synthetic a priori, mathematics, and not-self

Putting aside whether Kant has defined it as such or not but an "a priori statement" is but a basic assumption. In the context of a conventional system of thought of course the truth of such a statement may be necessarily assumed in order to safeguard the consistency of the system. However it is a matter of convention applied and therefore relative to the frame of reference.Pannapetar wrote:Now, there is a curious class of statements known to philosophers as a priori statements. Kant (who invented the term) has defined this as the class of statements that do not rely upon empirical verification. In other words, these are "logical" propositions that cannot be contradicted without violating logic.

Why does "C/2r=pi" describe the essence of a circle? What does "describe" mean in this context? A definition?Pannapetar wrote: For example, C/2r=pi describes the essence of the circle.

Kind regards

### Re: Synthetic a priori, mathematics, and not-self

I am not aware of any "mathematical laws".....can you explain one to me?

chownah

chownah

- Pannapetar
**Posts:**327**Joined:**Wed Jul 29, 2009 6:05 am**Location:**Chiang Mai, Thailand-
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### Re: Synthetic a priori, mathematics, and not-self

Yes, one thing leads to another. Take number theory, for example. Counting leads to the natural numbers. Attempting division on natural numbers lead to rational numbers; attempting to extract the root of natural numbers leads to irrational numbers, both lead to real numbers, and so on... One interesting philosophical question that has been discussed is whether mathematical properties do actually manifest in real world phenomena, or whether these properties are just ways of perceiving them, i.e. are purely mental constructs.gabrielbranbury wrote:I think numeric laws are simply a form of communication which correspond to phenomena. Without a circle there would be no pie.

I should probably have said "C/2r=pi describes the essential property of a circle" rather than it's "essence", because the essence (definition) of a circle is again a simple analytic statement: a line on which all points have the same distance from a given point p.TMingyur wrote:Why does "C/2r=pi" describe the essence of a circle? What does "describe" mean in this context? A definition?

Well, I tried to put this into simple layman's terms, so it simply means the sum of (correct) mathematical statements, equations, theorems, and axiomatic systems.chownah wrote:I am not aware of any "mathematical laws".....can you explain one to me?

Cheers, Thomas

- Prasadachitta
**Posts:**974**Joined:**Sat Jan 10, 2009 6:52 am**Location:**San Francisco (The Mission) Ca USA-
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### Re: Synthetic a priori, mathematics, and not-self

Hi Pannapeter,

Now Im at my shoeshine job waiting for the next shine.

I dont understand the question here. I dont see how properties of any kind can be anything but "real world phenomena". A nice thing to explore would be making an effort to see if we can distinguish "properties" from our "ways of perceiving them". I take this as a bit of a false construction which has instructional value but does not perfectly correspond to what is taking place.

I dont think C/2r=pi describes the essential property of a circle. A circle has many properties not described by "C/2r=pi". A circle has a certain aesthetic curvy quality which a person might remember but which only a circle can really convey.

Metta

Gabe

Now Im at my shoeshine job waiting for the next shine.

Pannapetar wrote: One interesting philosophical question that has been discussed is whether mathematical properties do actually manifest in real world phenomena, or whether these properties are just ways of perceiving them, i.e. are purely mental constructs.

I dont understand the question here. I dont see how properties of any kind can be anything but "real world phenomena". A nice thing to explore would be making an effort to see if we can distinguish "properties" from our "ways of perceiving them". I take this as a bit of a false construction which has instructional value but does not perfectly correspond to what is taking place.

[/quote]I should probably have said "C/2r=pi describes the essential property of a circle" rather than it's "essence", because the essence (definition) of a circle is again a simple analytic statement: a line on which all points have the same distance from a given point p.

I dont think C/2r=pi describes the essential property of a circle. A circle has many properties not described by "C/2r=pi". A circle has a certain aesthetic curvy quality which a person might remember but which only a circle can really convey.

Metta

Gabe

"Beautifully taught is the Lord's Dhamma, immediately apparent, timeless, of the nature of a personal invitation, progressive, to be attained by the wise, each for himself." Anguttara Nikaya V.332

### Re: Synthetic a priori, mathematics, and not-self

You wrote, "For example, C/2r=pi describes the essence of the circle. For all we know, it is eternal, universal, and non-changing."

(i) Math is merely a language of description, it doesn't exist "out there, essentially" and thus has no essence, nor can a shape described by such maths have such a thing. It is wholly convention, as all languages are, which means your questions are non sequitur.

(ii) For all we know... it is convention. It isn't eternal, nothing is: anicca. It isn't universal or non-changing for that reason as well. Math changes and maths die, general human history shows as much.

Your premise is unsound.

(i) Math is merely a language of description, it doesn't exist "out there, essentially" and thus has no essence, nor can a shape described by such maths have such a thing. It is wholly convention, as all languages are, which means your questions are non sequitur.

(ii) For all we know... it is convention. It isn't eternal, nothing is: anicca. It isn't universal or non-changing for that reason as well. Math changes and maths die, general human history shows as much.

Your premise is unsound.

### Re: Synthetic a priori, mathematics, and not-self

Well, it is debatable. Some would say that maths is a language of patterns as you say, and some that it is inherent in the world "out there" since not only does it provide a description but also pretty good predictions. Personally I side with the former. Maths is a result of our capacity for succinct logical codification of patterns we are able to perceive. So there is a lot of reference to the human being. To impute some absolute quality to it is not warranted to my way of seeing.Sobeh wrote:You wrote, "For example, C/2r=pi describes the essence of the circle. For all we know, it is eternal, universal, and non-changing."

(i) Math is merely a language of description, it doesn't exist "out there, essentially" and thus has no essence, nor can a shape described by such maths have such a thing. It is wholly convention, as all languages are, which means your questions are non sequitur.

(ii) For all we know... it is convention. It isn't eternal, nothing is: anicca. It isn't universal or non-changing for that reason as well. Math changes and maths die, general human history shows as much.

Your premise is unsound.

Still many mathematicians believe that they discover "laws" and chart the uncharted territories, rather than making things up. I think there is truth to both.

PS As for C/2r=pi describing the essence of the circle, this is a sloppy statement, since for it to make sense you have to already postulate what a circle is. Otherwise what is r? But I do appreciate the general point being made.

_/|\_

- Goofaholix
**Posts:**2964**Joined:**Sun Nov 15, 2009 3:49 am**Location:**New Zealand

### Re: Synthetic a priori, mathematics, and not-self

So are you trying to say that because a circle has an essence there must be an atman?

Whether or not a circle has an essence is besides the point, the circle itself is subject to the laws of impermenence, unsatisfactoriness, and not self.

When you experience something that you can define as a self let us know.

Whether or not a circle has an essence is besides the point, the circle itself is subject to the laws of impermenence, unsatisfactoriness, and not self.

When you experience something that you can define as a self let us know.

“Peace is within oneself to be found in the same place as agitation and suffering. It is not found in a forest or on a hilltop, nor is it given by a teacher. Where you experience suffering, you can also find freedom from suffering. Trying to run away from suffering is actually to run toward it.” ― Ajahn Chah

### Re: Synthetic a priori, mathematics, and not-self

Goofaholix,

I am not sure whom you are addressing but I for one would certainly not argue for a circle having any essence for reasons I've noted above.

I am not sure whom you are addressing but I for one would certainly not argue for a circle having any essence for reasons I've noted above.

_/|\_

- Goofaholix
**Posts:**2964**Joined:**Sun Nov 15, 2009 3:49 am**Location:**New Zealand

### Re: Synthetic a priori, mathematics, and not-self

I was replying to the OP, which I didn't really understand but heck what would I know I'm only in this for the end of suffering.Dan74 wrote:Goofaholix,

I am not sure whom you are addressing but I for one would certainly not argue for a circle having any essence for reasons I've noted above.

“Peace is within oneself to be found in the same place as agitation and suffering. It is not found in a forest or on a hilltop, nor is it given by a teacher. Where you experience suffering, you can also find freedom from suffering. Trying to run away from suffering is actually to run toward it.” ― Ajahn Chah

### Re: Synthetic a priori, mathematics, and not-self

Impermanence refers to form and phenomena, I think, not ideas and concepts. So I don't see anything resembling atman in mathematics (but Plato would disagree I guess). A circle is an abstraction derived by our intelligence from a variety of forms that share some characteristics and then codified using the laws of logic + geometry, again created by us, for the purpose of codifying patterns. Thus a circle has no more essence than the word "red".

I think the Buddha was onto another thing entirely when he spoke about impermanence and anatta. Although to many mathematicians their abstract universe is more real than the chair they sit on.

PS Maths is certainly no path to the ending of suffering - I see plenty of it around (and within). It can be a bit of an escape though!

I think the Buddha was onto another thing entirely when he spoke about impermanence and anatta. Although to many mathematicians their abstract universe is more real than the chair they sit on.

PS Maths is certainly no path to the ending of suffering - I see plenty of it around (and within). It can be a bit of an escape though!

_/|\_

- Modus.Ponens
**Posts:**2668**Joined:**Sat Jan 03, 2009 2:38 am**Location:**Gallifrey

### Re: Synthetic a priori, mathematics, and not-self

Are you a constructivist Dan74?Dan74 wrote:Impermanence refers to form and phenomena, I think, not ideas and concepts. So I don't see anything resembling atman in mathematics (but Plato would disagree I guess). A circle is an abstraction derived by our intelligence from a variety of forms that share some characteristics and then codified using the laws of logic + geometry, again created by us, for the purpose of codifying patterns. Thus a circle has no more essence than the word "red".

I think the Buddha was onto another thing entirely when he spoke about impermanence and anatta. Although to many mathematicians their abstract universe is more real than the chair they sit on.

PS Maths is certainly no path to the ending of suffering - I see plenty of it around (and within). It can be a bit of an escape though!

Just kiding. I don't like to discuss deep philosophical questions at forums since things are generaly discussed in unprecise terms and since philosophers discuss these things for years without coming to an end, so why bother. As an example, I know a book ("The Oxford Handbook of Philosophy of Mathematics and Logic ") that has more than 800 pages (writen by experts) on this subject and it still doesn't settle the discussion.

I myself am a platonist

He turns his mind away from those phenomena, and having done so, inclines his mind to the property of deathlessness: 'This is peace, this is exquisite — the resolution of all fabrications; the relinquishment of all acquisitions; the ending of craving; dispassion; cessation; Unbinding.'

(Jhana Sutta - Thanissaro Bhikkhu translation)

(Jhana Sutta - Thanissaro Bhikkhu translation)

### Re: Synthetic a priori, mathematics, and not-self

Thomas,

You delight in complication!

Sure it is true that a circle can be defined with an unwavering mathematical formula. So what? How do you get from there to Essence?

You delight in complication!

Sure it is true that a circle can be defined with an unwavering mathematical formula. So what? How do you get from there to Essence?

### Re: Synthetic a priori, mathematics, and not-self

How do you reconcile Platonism with Buddhadhamma? A belief in an ultimately real and unchanging world of forms seems to go directly against it (cf Hinduism).

_/|\_

### Re: Synthetic a priori, mathematics, and not-self

Essence has different meanings depending on context. A switch in the understanding of essence as a description of the mathematical basis for a circle to the idea that there are Atman-like qualities that we can know about phenomena is the

essence

of the mistake at the heart of the argument.

essence

of the mistake at the heart of the argument.

- Pannapetar
**Posts:**327**Joined:**Wed Jul 29, 2009 6:05 am**Location:**Chiang Mai, Thailand-
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### Re: Synthetic a priori, mathematics, and not-self

In some way it does. We observe entities and forces "out there" that manifest according to mathematical rules.Sobeh wrote:Math is merely a language of description, it doesn't exist "out there...

Mathematics is definitely not just a convention, because it is not arbitrary. Only the symbols and notation are arbitrary. Neither is a formula such as C/2r=pi or F = G (m1 * m2) / r^2 (Newtonian gravity) subject to impermanence (anicca). They remain unchanged for eternity. These formulas are as valid in a billion years as they are now and they are as valid in the Andromeda galaxy as they are here on Earth. Any intelligent alien species that might exist somewhere in the universe will discover exactly the same laws, although they might codify them in a different way.Sobeh wrote:For all we know... it is convention. It isn't eternal, nothing is: anicca.

How so?Goofaholix wrote:Whether or not a circle has an essence is besides the point, the circle itself is subject to the laws of impermenence, unsatisfactoriness, and not self.

Exactly my point. So could it then be said that concepts (in particular synthetic a priori aka eternal truth) are atman? What about the principle of consciousness that "sees" these eternal truths? I have no fixed opinion on this myself. It is puzzling and it is disregarded by the dhamma.Dan74 wrote:Impermanence refers to form and phenomena, I think, not ideas and concepts.

Very easy: essence is suchness, and the predicate C/2r=pi describes "circle"-ness. It applies to any circle-like object in Euclidean space.alan wrote:Sure it is true that a circle can be defined with an unwavering mathematical formula. So what? How do you get from there to Essence?

Cheers, Thomas

### Re: Synthetic a priori, mathematics, and not-self

Maybe you could ask this question to the nice folks over at Vedanta Wheel?

- Modus.Ponens
**Posts:**2668**Joined:**Sat Jan 03, 2009 2:38 am**Location:**Gallifrey

### Re: Synthetic a priori, mathematics, and not-self

To be honest I never dedicated myself that much to thinking in philosophy of mathematics and its relation to the dhamma. But anyway I can try a bit. Platonism comes in a variety of forms. I don't believe in a world of ideas hanging around waiting to be discovered. But I could never go to the other extreme and say that, for example, circles or prime numbers are mere constructions of our minds. Probably this is one of the questions to which the Buddha would remain silent if asked. I'll let the definitive answer come when I'm an arahat, whenever that may be. Until then I will identify myself loosely with platonism.Dan74 wrote:How do you reconcile Platonism with Buddhadhamma? A belief in an ultimately real and unchanging world of forms seems to go directly against it (cf Hinduism).

He turns his mind away from those phenomena, and having done so, inclines his mind to the property of deathlessness: 'This is peace, this is exquisite — the resolution of all fabrications; the relinquishment of all acquisitions; the ending of craving; dispassion; cessation; Unbinding.'

(Jhana Sutta - Thanissaro Bhikkhu translation)

(Jhana Sutta - Thanissaro Bhikkhu translation)

### Re: Synthetic a priori, mathematics, and not-self

Here it would probably pay to have some definitions handy. But even without these, how could any concept be atman if it is dependent upon cognizing (which is itself dependent upon other conditions)?Pannapetar wrote:Exactly my point. So could it then be said that concepts (in particular synthetic a priori aka eternal truth) are atman? What about the principle of consciousness that "sees" these eternal truths? I have no fixed opinion on this myself. It is puzzling and it is disregarded by the dhamma.Dan74 wrote:Impermanence refers to form and phenomena, I think, not ideas and concepts.

_/|\_

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