I don’t know how this will chalk up against politics and tanks, but…
I’ve always had deep personal concerns over the “existence” of numbers. To take it to the extreme I have doubted that pi, in fact, exists. My rationale?
Pi is transcendental, a non-repeating non-teminating decimal. That means that you cannot, for example, say something is pi units long unless you define it that way.
Truly, the same can be said of any number. So why the hell is math so powerful? I would not take the platonic view that numbers, as thoughts, have some meta-existence, but apart from that I am at a loss.
Well math is an abstraction, a mental representation of things that happen in nature. Its powerful because it describes what we perceive and allows us to employ models of reality. Without math you can’t have advanced science and technology. Math provides a workable abstraction.
I don’t know if I’m address your full concern, mostly because I’m not sure how to.
Not only can it be said of any number, but it can be said of anything. You can’t say “This is a chair” unless you define “chair” in a way to include that object. Does that mean that chairs don’t exist?
How many of you are there?
Mathematics is powerful because it allows us to speak precisely about a great many things. Pi, for example, is a lable for the ration of the circumference and diameter of any circle. We can, for instance, say that any circle is pi diamters around. Not the most useful of mathematical statements, when taken out of context, but quite powerful when applied in any number of formulations.
Mathematics is not a platonic set of ideals, it is a language designed to facilitate precise communication of certain sets of abstract ideas.
In other words, math is good because it works.
Of course, the existence of numbers depends on your definition of “existence”; you can’t see, feel, touch, or heal a number, but counting numbers, for instance, are abstract concepts that describe the real world. You can talk about 3 balls, 7 computers, what have you, and it is clear that you are talking about quantity, and defining what that quantity is. In that sense, counting numbers certainly exist as ideas that have meaning in our physical world. What would it mean to say that I have “ten” fingers if the number “ten” did not exist, even as an idea?
From there, there is a very natural progression in the development of the integers, the rationals, the algebraic, real, and complex numbers. All of these, too, have various realizations in the physical world describing quantity or measurement.
Take pi, for example. Pi is the ratio of the circumference of a circle to its diameter; throughout our environment we immediately see instances of naturally occuring circles (or physical world approximations to perfect circles, anyway)–the shape of the sun or moon, ripples on a pond, the shapes of bubbles, for simple examples. Pi has been studied for thousands of years, and it was quite surprising when it was discovered to be transcendental. We didn’t define pi’s nature ourselves, we merely noted it’s realization in the physical world (in the form of circles), studied it, and came upon these discoveries. We didn’t force these properties on pi, they were always properties of pi; in fact, conversely, we were forced to accept theses properties of pi, regardless of how unexpected and surprising they may be. That seems to me a pretty strong argument for the existence of pi as a fundamental and unalterable abstract concept.
A related paper describing some of this in more detail, kind of longish, but worth reading, on this topic, is linked here. This paper is kind of an extension of a previous paper on the same topic, linked here, also worth reading; however, specific to the OP I believe the former is more directly relevant.
There are many mathematicians who maintain that numbers and mathematics exist regardless of whether humans (or sentient beings) are alive to recognize it. Such a debate skirts perilously close, if not entirely into the territory of;
If a tree falls in the forest and nobody is around to hear
[sup](it, do all the other trees point and laugh?)[/sup]
it, does it make any sound?
As a proponent of the transcendental validity of science and logic, I must concur that it is very possible that primal mathematics may be as much a fact of reality as quantum mechanics or stars.
I will refrain from attempting any vague or rambling discourse upon this notion. Instead I shall leave you with a saying that elicits a distinct note from “The Music of the Spheres.”
Mathematics is music for the mind,
music is mathematics for the soul.*
This debate reminded me of a passage in a work of Friedrich Engels called Anti-Dühring - a defense of materialist philosophy - that dealt exactly with this question:
Essentially pure mathematics cannot exist without the human brain to create the abstractions of mathematics in the first place. Circles and their circumferences and their diameters are all physical manifestations, but the concept of the ratio between the latter two exists only because there were people around to think it up.
There’s a fundamental difference between “does pi exist if there’s nobody around to conceptualize it” and the old chestnut about a tree falling in the forest. The tree example deals entirely with physical manifestations - the disruption of the wood as it splits, the impact of the tree on the ground under it, and the vibrations created by displacement of the air because of these events. So a tree falling in a forest under any circumstances makes a sound.
Determining the existence of pi, on the other hand, involves more than just labeling physical manifestations. It is more than saying “This is a circle, this is the distance around its edge, and this is the distance between two points opposite each other on the edge.” First, it involves stripping the distances of all properties except one - the numbers representing them. Then it involves applying the concept of division to the two numbers and coming up with a third. Neither of these processes are physical manifestations of the real world and thus do not have an existence independent of the human mind.
I missed an important part of the quote. Please forgive the mess, but it’s important to the argument I was making.