Is mathematics independent of human consciousness?

I'm strongly inclined to say yes . Here's an argument. If there's even one technological civilization elsewhere in our unimaginably vast universe, then that civilization must have discovered enough math to produce technology. But we have no reason at all to think that it's a human civilization, given the very different conditions in which it evolved: if it exists, it belongs to a different species from ours. So: If math depends on human consciousness, then we're the only technological civilization in the universe, which seems very unlikely to me. Here's a second argument. Before human beings came on the scene, did the earth orbit the sun in an ellipse, with the sun at one focus? Surely it did. (Indeed, there's every reason to think that the earth traced an elliptical orbit before any life at all emerged on it.) But "orbiting in an ellipse with the sun at one focus" is a precise mathematical description of the earth's behavior, a description that held true long before consciousness emerged here....

In writing mathematical proofs, I've been struck that direct proofs often seem to offer a kind of explanation for the theorem in question; an answer the question, "Why is this true?", as it were. By contrast, proofs by contradiction or indirect proofs often seem to lack this explanatory element, even if they they work just as well to prove the theorem. The thing is, I'm not sure it really makes sense to talk of mathematical "explanations." In science, explanations usually seem to involve finding some kind of mechanism behind a particular phenomenon or observation. But it isn't clear that anything similar happens in math. To take the opposing view, it seems plausible to suppose that all we can really talk about in math is logical entailment. And so, if both a direct and an indirect proof entail the theorem in question, it's a mistake to think that the former is giving us something that the latter is not. Do the panelists have any insight into this?

You've asked a terrific question! I wish I were more qualified to venture an answer to it. As you suggest, a sound direct proof of a theorem shows that the theorem must be true, in the broadest possible sense of "must." But a sound indirect proof shows the same thing. The difference, if any, seems purely psychological: some people find one proof psychologically more satisfying than the other. My sense is that some philosophers of math take this psychological difference very seriously and propose far-reaching revisions to classical math on the basis of it. You might take a look at the SEP entry on intutionism in the philosophy of math , particularly the discussion of constructive and nonconstructive proofs. The entry includes other helpful links and references too.

Imagine that a Greek philosopher promised to his queen that he would determine the greatest prime number. He failed. Do you think that the mathematical fact that primes are infinite was a cause of his failure? I'm asking this because I guess most philosophers think that mathematical facts have no causal effects.

You've asked an interesting question, one related to what's often called the "Benacerraf problem" in the philosophy of mathematics (see section 3.4 of this SEP entry ). I'm not sure that the problem is peculiar to mathematics. Imagine that the philosopher tried to impress his queen by creating a colorless red object. Was his failure caused by the fact that colorless red objects are impossible? If facts about color and facts about redness in particular can have causal power, can the fact that colorless red objects are impossible have causal power? Part of the problem may be that these questions assume that we have a better philosophical grasp of the concept of fact and the concept of cause than we actually do. Given our currently poor grasp of those concepts, I don't think we should be confident that mathematical explanations or mathematical knowledge must depend on the causal power of mathematical facts.

My understanding is that we can use systems like Peano Arithmetic to prove the seemingly basic truth that 1+1=2. Do such proofs actually give us reasons to believe that 1+1=2 that we didn't have before? Are they more fundamental or compelling than whatever justification a mathematically-naive person would have to believe that 1+1=2?

There are genuine philosophers of math on the Panel, but while we wait for them to respond I'll take a stab at your questions, which are epistemological as much as they're mathematical. I think we can answer yes to the first question without having to answer yes to the second question, but the answer to both questions may be yes . As I understand the Peano Proof that 1 + 1 = 2 , the gist is that the definitions of 'successor', 'addition', and '2' imply that 1 + 1 = 2. The successor of 1 is defined as 2, and addition is defined so that the result of adding 1 to any number is the successor of that number. Therefore, the result of adding 1 to 1 is 2. If the Peano Proof constitutes a reason to believe that 1 + 1 = 2, then it's surely a reason we didn't have before we had the Peano Proof. So I (somewhat tentatively) answer yes to your first question, regardless of the answer to your second question. Even if we grant the infallibility of the deductive inferences in the Peano Proof, the...

I am interested in how mathematical propositions relate to objects in the world; that is, how math and its concepts somehow correspond to the physical world. I have thought a bit about the issue, and realize that what happens, say, with numbers when we do some kind of mathematical operation with them may be the same as when we deduce one proposition in logic from another (If there is a number 2 and an operation "+", and an operation "=", then the result of using 2 + 2 = 4); but my question is this: does the proposition 2 + 2 = 4 mean the same thing as taking two objects and placing two more objects alongside of them, and then counting that there are four objects?

Philosophers continue to debate the relationship of mathematics to the physical world, including why mathematics is so effective at describing the physical world. The SEP entry on "Explanation in Mathematics," available at this link , contains much useful discussion as well as many references to further reading. At least one of the articles cited in the bibliography is available online: The Miracle of Applied Mathematics , by Mark Colyvan. I hope these prove helpful. Strictly speaking, the proposition that 2 + 2 = 4 can't mean the same thing as the process of taking two objects, placing two more objects alongside them, and then counting that there are four objects in total. Propositions and processes belong to different categories. Moreover, one might doubt that the proposition that 2 + 2 = 4 even entails that whenever you take two physical objects and place two more physical objects alongside them, there will be four physical objects to count up. Why?...

Does the fact that our perceptions can be represented geometrically and that geometry consists of eternal truths independent of the mind prove that an external reality underlies our perceptions?

I don't think that such an argument would rationally compel external-world skeptics (who say that no one can know that there's an external world) to abandon their view. External-world skeptics think that no one can know that solipsism is false, where solipsism is the claim that nothing external to oneself and one's mind exists. The solipsist won't grant that geometry consists of truths that are independent of his own mind, because he thinks nothing is. The solipsist could admit that his perceptions have a geometric character to them without having to attribute that character to something external. So I don't think solipsism can be disproven in the way you suggest. All of this assumes that solipsism is otherwise intelligible. But one might argue that solipsism is unintelligible because it relies on the incoherent idea of a 'private language', an idea explored in detail in this SEP article .

In mathematics numbers are abstract notions. But when we divide number say we do 1 divided by 2 i.e. ½ does this mean abstract notions are divisible. It gives me a feeling like abstract notions have magnitude but then it comes to my mind that abstract has no magnitude.1=1/2 + 1/2 can we say the abstract notion 1 is equal to the sum of two equal half abstract notions? How should I conceptualize the division? The other part related to abstract notion is that how is the abstract notion of number 1 different from the unit cm? how can we say that the unit cm is abstract when we consider it a definite length. How is the unit apple different from unit cm if I count apples and measure length respectively? I am in a fix kindly help me to sort out this. I will be highly- highly grateful to you.

You asked, "Does this mean that [these particular] abstract notions are divisible?" I'd say yes . But that doesn't mean they're physically divisible; instead, they're numerically divisible. Abstract objects have no physical magnitude, but that doesn't mean they can't have numerical magnitude. The key is not to insist that all addition, subtraction, division, etc., must be physical. I'd say that the number 1 (an abstract object) is different from the cm (a unit of measure) in that the cm depends for its existence on the existence of a physical metric standard: for example, a metal bar housed in Paris or the distance traveled by light in a particular fraction of a second (where "second" is defined in terms of the radiation of a particular isotope of some element). In a universe with no physical standards, there's no such thing as the cm and nothing has any length in cm. By contrast, the number 1 doesn't depend for its existence on anything physical. Apples are physical, material objects. Units...

How would a philosopher of math describe what happened when ancient mathematicians discovered (?) the number zero?

I think the answer will depend on which philosopher of math you ask. As you seem to recognize, some philosophers of math deny that numbers exist independently of us in such a way that their existence is genuinely discovered by us. Even philosophers of math who think that numbers are discovered might say that your question -- "What happened?" -- is an empirical historical or psychological question rather than a philosophical one. In any case, you'll find relevant material in the SEP entry on "Philosophy of Mathematics" at this link .

Does a point in geometry (cartesian and euclidean) occupy space or have volume (if we consider 3-D geometry)? And is a line segment always perpendicular to its point of origin? Or can we frame this as, is a line perpendicular to each and every point lying on it?

As I understand the theory, an individual point in geometry has no extension and no volume; it's in space but doesn't occupy space in the sense of taking up a nonzero amount of space. Being perpendicular is a relation between lines (or line segments) rather than a relation between a line (or a line segment) and a point. A point can't be perpendicular to anything. At any rate, there's no more reason to say that a line is perpendicular to each point lying on it than to say that it's parallel to each point lying on it. I think it's neither.