# Need Help: How is arithmetic synthetic ?

• foxtrot_echo_
In summary: Foundations_of_Arithmetic_(Frege)...and you will see how Frege developed his axioms from a small set of logical axioms (the ones we now call the propositional calculus). The key point is that this development is completely deductive, so there is no empirical content. Thus, arithmetic is analytic, not synthetic.In summary, Kant's error was to think that the subject matter of arithmetic was quantity. But it isn't. The subject matter is SYSTEMS OF QUANTITIES or, as we say today, STRUCTURES. And structures are logical entities, not empirical ones.In summary, modern philosophers have differing opinions on whether the statement "7 +
foxtrot_echo_
I am having difficulty in understanding how the statement "7 + 5 = 12 " is a synthetic statement ?

(I don't know whether this view is commonly accepted or whether opinion is divided on this issue.But I would like to know arguments put by those in favor of it.)

I didn't have trouble in understanding that a statement such as "A bald man is a man" is analytic .

But I am having trouble understanding why the statement "7 + 5 = 12" is not analytic.

Can someone explain or provide references where it is explained (it would be helpful if the source is self-contained as I am a beginner).

(it would be great if the explanation is in light of modern philosophy).Thanks in advance!

This may help...

http://en.wikipedia.org/wiki/Philosophy_of_mathematics#Empiricism

"Empiricism is a form of realism that denies that mathematics can be known a priori at all. It says that we discover mathematical facts by empirical research, just like facts in any of the other sciences. It is not one of the classical three positions advocated in the early 20th century, but primarily arose in the middle of the century. However, an important early proponent of a view like this was John Stuart Mill. Mill's view was widely criticized, because it makes statements like "2 + 2 = 4" come out as uncertain, contingent truths, which we can only learn by observing instances of two pairs coming together and forming a quartet."

Thanks for the links. I would like to verify how far I have understood the issue.
Can the following be used as a valid argument in proving that arithmetic is a priori synthetic :-

Consider a statement such as "A bald man is bald". Here we can deduce the "bald" in predicate from the "bald" in subject only .
The analysis doesn't depend on "man" at all.
As another ex :-
"A bald antelope is bald" is also true.
"A bald tiger is bald" is also true.

But same can't be said in arithmetic.
"7+5 = 12" but "7+6 = 13" . Thus the "12" in first case and "13" in second can't be deduced from "7" alone.
We need the entire complex whole consisting of "7","+","5" in the first case to deduce "12".
Similarly,We need the entire complex whole consisting of "7","+","6" in the first case to deduce "13".

I guess, this must be a crude amateurish argument to prove that arithmetic is a priori synthetic,but is my understanding of the issue somewhat on the right track.

P.S - a different query , I read about this issue in The Problems of Philosophy(1912) by Bertrand Russell. Has the book been reviewed on physicsforums in book discussion? If yes can you provide the link.

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Whether or not synthetic a priori knowledge is possible is an open question. Rationalists argue that nothing can be synthetic a priori. Kant famously used a "transcendental argument" to show that "7 + 5 = 12" is synthetic a priori. See http://www.iep.utm.edu/kantmeta/#H3. The best way to understand the argument is to read it yourself. Do a search for "7 + 5" in http://www.e-text.org/text/Kant%20Immanuel%20-%20The%20Critique%20of%20Pure%20Reason.pdf.

But not everyone agrees. Kant was responding to the rationalism of other philosophers of his time. See also:

http://www.britannica.com/EBchecked/topic/578646/synthetic-a-priori-proposition
http://plato.stanford.edu/entries/rationalism-empiricism/
http://plato.stanford.edu/entries/analytic-synthetic/

Yes, I read the transcedental argument of Kant.

But I would also like to know what proof the modern day philosophers who are proponents of this theory use.

I may be coming at this from a different angle than you are looking for, but doesn't "synthetic" refer to the fact that "7" and "5" are empirically observed quantities and the sum, "12," is synthesized by processing the data at an abstract level? If the "12" would be directly observed, would it be a thesis instead of a synthesis?

foxtrot_echo_ said:
Yes, I read the transcedental argument of Kant.

But I would also like to know what proof the modern day philosophers who are proponents of this theory use.

In what I've read at least, modern day philosophers seem more interested in other issues . It is interesting, though, that modern modal arguments using "possible worlds," which are very common, seem to presume empiricism over rationalism. I can't recall any arguments specifically given to reinforce this usage, but they probably exist.

Thanks for the replies . I will check those links.

Anyways , my earlier understanding was flawed , wasn't it :-

foxtrot_echo_ said:
Consider a statement such as "A bald man is bald". Here we can deduce the "bald" in predicate from the "bald" in subject only .
The analysis doesn't depend on "man" at all.
As another ex :-
"A bald antelope is bald" is also true.
"A bald tiger is bald" is also true.

But same can't be said in arithmetic.
"7+5 = 12" but "7+6 = 13" . Thus the "12" in first case and "13" in second can't be deduced from "7" alone.
We need the entire complex whole consisting of "7","+","5" in the first case to deduce "12".
Similarly,We need the entire complex whole consisting of "7","+","6" in the first case to deduce "13".

I guess I was using "7" as a subject whereas "7+5" in its entirety is the subject.

Hi Foxtrot Echo,

If we had to put it in Kantian terms, we would have to say that all arithmetical knowledge (even so simple a proposition as '1 + 1 = 2') is most emphatically SYNTHETIC A POSTERIORI.

Kant's ideas on the epistemological status of arithmetic were seriously flawed because, in his day, there was no satisfactory statement of the axioms of arithmetic. This did not come until the late 19th C. If you have the opportunity, read the first few chapters of Russell's "Mathematical Philosophy" for a very interesting and readable explanation of Peano's work in this field.

Two things might strike you about Peano's axioms. Firstly, there is nothing intuitive or self-evident about them. This shouldn't be surprising, of course, because it took two and a half millennia and an awful lot of intellectual sweat before somebody finally worked out what they are! Secondly, the cardinal numbers are not part of the axioms, but have to be created, or stipulated, as a separate act of definition (just as Euclid begins the Elements by stipulating definitions of essential terms like "line" and "circle").

It follows, we have to concede that 1 + 1 = 2 is only contingently true. If we wanted to develop an alternative arithmetic, analogous with non-euclidean geometry, we need only negate or amend anyone of the axioms to arrive at a different result.

Everything Kant had to say about mathematics, and the perception of space and time in general, was comprehensively shot down within a couple of decades of his death, when the first papers were published proving the validity of non-euclidean geometry. Of course, very few philosophers realized it at the time, and I think some still don't, even now...

Like Aristotle's 'essence' and 'attribute', and Plato's Theory of Forms, Kant's eptistemological categories (a priori, analytic etc) have been abandoned by the sciences as a museum-piece, a staging-post on the road to knowledge.

## 1. What is arithmetic synthetic?

Arithmetic synthetic is a method used to perform calculations in mathematics without actually using numbers. It involves using symbols and variables to represent quantities and then manipulating them to find a solution.

## 2. How is arithmetic synthetic different from traditional arithmetic?

The main difference between arithmetic synthetic and traditional arithmetic is that synthetic arithmetic uses symbols and variables instead of numbers. This allows for a more general and abstract approach to solving problems, whereas traditional arithmetic relies on specific numerical values.

## 3. What are the benefits of using arithmetic synthetic?

One of the main benefits of using arithmetic synthetic is that it allows for a more efficient and organized way of solving problems. It also helps to develop critical thinking and problem-solving skills, as it requires a deeper understanding of mathematical concepts.

## 4. Can arithmetic synthetic be used for more complex calculations?

Yes, arithmetic synthetic can be used for more complex calculations, such as solving equations with multiple variables. However, it may require more advanced techniques and knowledge of algebraic principles.

## 5. How can I improve my skills in arithmetic synthetic?

The best way to improve your skills in arithmetic synthetic is through practice. Start with simple problems and gradually work your way towards more complex ones. It is also helpful to study and understand the underlying principles and rules of synthetic arithmetic.

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