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Math and Geometry are synthetic a priori?!

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» Math and Geometry are synthetic a priori?!

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For those who already know about the different propositions skip to the dotted line.

For those who don't know, Kant proposed that there are there are two types of proposition, a priori (before experience) and a posteriori (after experience) and then two further subcategories, analytic and synthetic.

Analytic means that the predicate is contained in the notion. It tells you nothing more and you do not need experience outside the statement to be able to say whether it is true or false, e.g. "A triangle is a three sided shape". The very meaning of triangle is a three sided shape. So all analytic statements are unfalsifiable.

Synthetic means you add something new and need experience outside the statement in order to say whether or not its true. e.g. "This cat is black". We don't know until we see the cat. Synthetic statements are falsifiable.

Note, before Kant people such as Leibniz and Hume said (or would do in Kant's terms) you can only have synthetic a posteriori propositions and analytic a priori ones. However, Kant's grand finale of all this explanation was to say that you can have synthetic a priori. He said that metaphysical claims were exactly these kind.

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Now, Kant also said that mathematical and geometrical propositions are synthetic a priori, and this is where I have a problem. He says "7+5=12" is not analytic because our intuition is used to come to the subject (12) from the predicate ("the sum of 7 and 5"). Also in geometry, "A straight line is the fastest route", because the idea of the fastest route is not contained within the idea of a straight line.

But surely on this thinking the statement which was our example of an analytical a priori claim, "A triangle is a three sided shape" needs our intuition. The idea of a three sided shape being contained within the idea of triangle is the same as the idea of 12 being contained within the idea of "the sum of 7 and 5", no?

Am I missing something?

@markoos,

im probably missing alot... but I tend to think its not healthy to try to categorize and lump everything... i mean is this totally irrelevant but doesnt opinion and/or logic have any pull on stuff like this... OK yeah a triangle is a three sided shape but TECHNICALLY is there such thing as a 2d triangle whether isoceles scalene or equilateral? I mean dont we live in 3 dimensions and no matter how slight, things have a depth, width and height... so what would you say about a piece of pizza... is it triangular even though the outside is a curve? I dunno... I just mean to challenge the purpose of categorizing triangles like that... what good does it truly do...

12 can be made of 11+1, or even 15-3.... so i mean why bother guessing at stuff like this and calling it analytic or synthetic... **** itd probably do good for them to add a few categories like anasyn for stuff like when the fastest route is going a straight line until you find out theres construction in your way....
seriously what good is the fastest route is a straightline when theres just no way to travel straight. I mean even in airplanes it can be faster to go slightly off course and ride the jet stream...

ok i know im weird and im not stickin to the priori thing... but it aggrivates me when things have to be this or that.. and not just be or even at least just be circumstantial

what would emotion be?
analytic or synthetic...

and isnt EVERYTHING falsifiable...

what if i said hey i drew a triangle! and your like ok thats analytic... but then you look at my paper and seen i drew a square and must of had a horrible education... UH OH the analytic has been falsified!

@markoos,

markoos wrote:

But surely on this thinking the statement which was our example of an analytical a priori claim, "A triangle is a three sided shape" needs our intuition. The idea of a three sided shape being contained within the idea of triangle is the same as the idea of 12 being contained within the idea of "the sum of 7 and 5", no?

Well, I'm not sure, but the thinking may be as follows.

A triangle is already defined as a three-sided shape, so it does not need any intuition to get from 'triangle' to 'three-sided shape'. But 7 + 5 is not defined

@markoos,

Can't it also be a posteriori in some ways, such as the old primary school problems of 'I have 3 oranges in a box, and add 5 more oranges. How many oranges are in the box?' We learn from the existence of the oranges and the total in the box.
(granted this post makes me feel 'simple' among the other ones)

@Kolbe,

No, I think that would be a priori (but synthetic). As I understand it, 'a priori' does not necessarily mean that we apprehend the answer immediately; it means that, once we have apprehended it, we realise (or should do!) that it will be true in all similar cases. There will be no odd counter-examples, as there always may be for a posteriori statements e.g. 'no person has more than two hands'.

@markoos,

i know this is a little old but this forum came up on google search - i was wondering if someone could explain what role contradictions played in maths being defined as synthetic a priori.
i know hume would say maths was analytic a priori - what's wrong with this?
also if by demonstration i were to realise a straight line is the shortest distance between point A and point B on a flat plane, and then realise that the shortest distance between two points must be a straight line, but then look at a curved surface and realise that my supposed 'rule of mathematics' was in fact not necessarily true could maths then still be synthetic a priori? dont you have to look at different situations to understand which rules make sense when?

@treasureisland,

treasureisland;161671 wrote:

i know this is a little old but this forum came up on google search - i was wondering if someone could explain what role contradictions played in maths being defined as synthetic a priori.
i know hume would say maths was analytic a priori - what's wrong with this?
also if by demonstration i were to realise a straight line is the shortest distance between point A and point B on a flat plane, and then realise that the shortest distance between two points must be a straight line, but then look at a curved surface and realise that my supposed 'rule of mathematics' was in fact not necessarily true could maths then still be synthetic a priori? dont you have to look at different situations to understand which rules make sense when?

Excellent question. Mathematicians are still debating this sort of question. (Incidentally, a straight line on a sphere is still the shortest line, but this line would have to penetrate the sphere. Which is fine in theory but not for airplanes and battleships.)

I don't think there's a simple answer to your question. It's a rich subject. I recommend looking into Godel, Turing, Chaitin, and Incompleteness. I personally feel that math is rooted on a certain amount of intuition but quickly is forced to move beyond this. Why is 0! equal to 1? Because it's convenient. Why is 1 not a prime number? Because it's convenient for certain theorems. In what way to transcendental numbers exist? Does a perfect equilateral triangle exist only within our mind? Kronecker is a fascinating guy, but he's hard to research. I can only find fragments of his thought, unfortunately. And yet I could probably find what Brittany Spear's favorite food was a a child....what a world!

@markoos,

markoos;41753 wrote:

Now, Kant also said that mathematical and geometrical propositions are synthetic a priori, and this is where I have a problem. He says "7+5=12" is not analytic because our intuition is used to come to the subject (12) from the predicate ("the sum of 7 and 5"). Also in geometry, "A straight line is the fastest route", because the idea of the fastest route is not contained within the idea of a straight line.

But surely on this thinking the statement which was our example of an analytical a priori claim, "A triangle is a three sided shape" needs our intuition. The idea of a three sided shape being contained within the idea of triangle is the same as the idea of 12 being contained within the idea of "the sum of 7 and 5", no?

Am I missing something?

Three sided polygon defines triangle.
7+5 does not define 12.

@markoos,

markoos;41753 wrote:

---------

Now, Kant also said that mathematical and geometrical propositions are synthetic a priori, and this is where I have a problem. He says "7+5=12" is not analytic because our intuition is used to come to the subject (12) from the predicate ("the sum of 7 and 5"). Also in geometry, "A straight line is the fastest route", because the idea of the fastest route is not contained within the idea of a straight line.

But surely on this thinking the statement which was our example of an analytical a priori claim, "A triangle is a three sided shape" needs our intuition. The idea of a three sided shape being contained within the idea of triangle is the same as the idea of 12 being contained within the idea of "the sum of 7 and 5", no?

Am I missing something?

I agree with you. Though I think this is easily "saved" if you argue that the triangle was a priori synthesized, and then due to that synthesis you could then look at it analytically (Sides being made of lines) thereby looking at it as an pedagogical example as opposed to a definitional one.

"A triangle has three sides", clearly provides the concept within the statement. With 7+5 , the number 12 is not in the statement, and must be conceived. Kant's argument for the Synthetic priori of numbers is, that we have the intuitive faculty of being able to synthesize the concept of 12; that "7 things" "Plus" "5 things" equals "12 things" (note that the numbers 7,5,12 and the 'plus sign' are not even prerequisites for this faculty).

12 is related to 7 and 5 in a certain way that is not explicitly contained within a statement, and we have a natural faculty of being able to visually see the sum.
A few interpretations of Kant's synthetic a priori, is supported even by modern data:

The results of Woodcock Johnson-III radex analysis and the Woodcock Johnson-R correlation matrix, stand against the notion that math ability is a crystallized (knowledge) factor. It's noteworthy that the WJ-III applied problems is highly g-loaded (exceeding even concept formation). But additionally, the WJ-R correlation matrix reveals the higher relationships between applied problems (and quantitative concepts), and clearly non-crystallized abilities ,such as sound-blending and spatial abilities. Those types of correlations would not exist had applied problems been reflecting learned abilities. The natural assumption is that applied problems and quantitative concepts performance reflects differences in mathematical thinking at the axiomatic level (though this doesn't imply that it is entirely untrainable). The most appreciable thing about the CoGAT (in spite of it's low overall validity), is the inclusion of the 'equation balancing' task, which reflects our handling of mathematics near the Peano axiom level.

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