Mathematical systems are an excellent model for organizing and conducting thought:
In the mathematics community, any argument in support of a conjecture, that deviates from "sound argument" never flies except by mistake.
Why don't we do the same type of organization and hold ourselves to the same standard?
Math proof = Sound argument
We'd likely see a lot corollaries of conjectures but that's okay if we're in this for the long haul.
Do you have a proof that we don't hold ourselves to higher standards?
There's actually a rather interesting little corner of mathematics called "proof theory." It deals with the question of what a proof is and how can we use them. It starts to look like philosophy from time to time.
I think the real difference is that mathematics typically starts with a small set of very crisply defined ideas, and then manipulates them. A lot of philosophy comes from dealing with ideas that get murky when you look at them too long. The tools of proof theory are not designed to work with that.
If I'm understanding your question correctly, then you're basically asking "why doesn't philosophy have the same level of rigor as mathematical proof?"
I think there's two parts involved in answering this.
First, one aspect of philosophy for many philosophers (arguably all) is that philosophy is actually a form of history, meaning we are studying ideas that someone else wrote about and discussing what is that they wrote and meant. This almost definitionally means there will be interpretive disagreements. (Contemporary analytic philosophy is no way exempt from continuing to do history).
Second, rigor seems nice until you realize rigor is double-sided. The smaller the set of operators and values, the more rigor you can obtain, but the price is That the size of its domain is automatically smaller than what you can say or do with less rigor.
For instance, how would you discuss with a mathematical proof whether mathematical proofs are sufficiently rigorous?
Philosophical theories are more like scientific theories than mathematical theories, in that they have empirical content. As such, there aren't any (universally agreed upon) "first principles" that must be respected. Any potential first principles might get discarded if the reasons for doing so are compelling enough. And even if there are some such principles (say, some nearly logical principles or the principle of sufficient reason) they wouldn't be strong enough to answer every interesting philosophical question that we might ask. We have to make "guesses" and test these guesses against our data and our intuitions. And making arguments as precise as mathematics requires solidifying our concepts with rigid definitions; but usually concepts are fluid and open to revision in light of new data. We don't always know what we're talking about at the beginning, so we give something a name and work towards a gradual understanding of it, rather than having a complete definition worked out from the start. (These points such as "no obvious first principles", no "rigid concepts", "gradual understanding", hold for science too, since you mentioned scientific communities in your question.)
I'm sympathetic to a formal approach myself, and I'm a metaphysical realist. To me this means, at a minimum, that there exists mind independent objects that bear real properties and stand in real relations to other objects; the world has structure. Our theories also have structure, and when the the structure of our theories match up with the structure of the world (our nouns name objects, our predicates name properties and relations in a structure-preserving ("isomorphic") way), it's true. But, merely formalizing our philosophical concepts wouldn't do much good in determining whether they match up to real objects/properties/relations, ie. whether some proposed theoretical objects are real or statements about them are true. For example, saying something about causation and then formalizing it wouldn't do much good in determining if our formal notion of causation corresponds to a real relation between events (or whether there even are real objects such as events). To decide that, we need to go out into the world and gather data.
There are some famous "formal proofs" in philosophy such as Gödel's ontological argument, but formalizing it doesn't make it true. Finding an interpretation of Gödel's "positive property" notion is not that easy, especially one that corresponds to all and only the divine attributes.
A quick Google search also turned this neat project up: A formalization of Spinoza's Ethics, Part 1: Consequences for interpretation. I admit I didn't read it, but the point is that just because Spinoza is made more precise and mathematical, it doesn't make it any more true. (It may be true, and formalizing might make it easy to understand or settle any ambiguities that might come up, but there's still the question about correspondence to reality).
A proof is only as strong as the axioms it is built upon. Mathematics works over a very limited number of strong axioms to work with, which gives it a limited number* of things that can be proven, but the proofs are very strong thanks to the axioms they work with (and prior proofs relying on the same axioms).
Philosophy works with much broader field of ideas and does not always have strong axioms to anchor the thoughts in. This is not a mistake, nor a weakness, but a tradeoff. The broader is the scope of your work, the more likely you will have to settle for ideas that are not proven to be absolute fact.
Remember that the fact that an idea is not an absolute 100% true fact is not always a problem, provided it works well for building a construct. To give you math/physics analogy, Newtonian motion is known to be inaccurate, yet it's short of being really wrong. We still use it and it's quite handy.
It would be a shame to give up a thought that could reveal some important truth just because it's not absolute 100% verified truth.
* The limited number is infinity, but still a limited one much like the set of all even numbers.
Because it would then cease to be philosophy.
Philosophy sees itself as the progenitor of all the sciences, as its questions lead to the paradigm shifts upon which branches of science are founded. To limit itself to a predetermined set of rules would be to strip itself of the flexibility needed to come up with the next new thing.
In other words, it is precisely because it does not follow axioms and proofs of a specific scientific paradigm that it is able to come up with new ones.
Often, when a useful higher standard for arguments is discovered, a brand new field of study is created based on that higher standard, or maybe a subfield/subtopic of an existing field of study. For example, the study of physical sciences replaced the study of natural philosophy, and the study of the infinite is a mathematical topic.
Since the arguments being held to a higher standard tend to fall into these other fields, the things we still call philosophy are much more likely not to be held to such.
Another thing I would add is that proofs are built on strong axioms, but also on precise definitions.
It's hard to find a precise and universally accepted definition for any complex concept in philosophy.
Those are a much harder to define than a point, a circle or a function (not that they're easy either)
You can't build any sort of proof if everyone doesn't agree on a definition. And your can make you own, but anyone can reject your conclusion if they don't agree to your axioms or definitions.
Spinoza tried a system of axioms for his Ethics, if you want to what it looks like.
What is stopping the philosophical community from holding themselves to the same standard?
The impression that the philosophers' "standards" are not sufficiently high, I think, is due to (1) the apparent lack of progress in solving philosophical puzzles in conjunction with (2) the deceiving simplicity of these puzzles.
In fact, nothing stops the philosophical community from achieving clarity, rigor and conclusivity. A systematic application of sophisticated logical tools to common philosophical problems over the course of the past century have lead to an enormous transformation in our ways of handling them. At least some "classical problems of philosophy" (the problem of universals, for example) have been solved. At the same time, the "standards" really have become much higher. Overall, it may seem that philosophers do not really advance — only because the advancement is piecemeal and unrushed.
The impression that philosophical problems are "easy" stems from the very high general intelligibility of the words used to state them. In fact, there are terribly complex — they are about thinking itself. However, consider the Fermat Last Theorem, which statement sounds incredibly simple. Still, it took 358 years to establish that there is indeed no three integers a, b, с that satisfy a^n + b^n = c^n if n is integer and greater than two. There is no reason to think that philosophical problems can be solved any faster — they deal not with the properties of relatively simple objects, but with the thinking itself.
The mathematical community has proofs
Please note that some of the best practitioners of mathematics disparage proofs. Lefzchetz, for example, told his students not to just present pretty new proofs (they probably already knew that they ought not to present ugly ones!). He wanted new substantiatial new ideas. Poincare was similarly disparaging in his book Geometry and the Imagination, where he placed the mathematical imagination - that is induction, most broadly understood (it ought not to be confused with mathematical induction) on a higher plane than that of deduction - the bread and butter of proofs.
Different communities of thinking have different ways of assessing ideas. This is why philosophical argument does not look like mathematical argument. They are addressing different questions with different standards of argument - that is reasoning - and not proof.
The mistake here, I think, is that the question assumes that philosophy is simply another field of research on par with mathematics, physics, or whatnot. But philosophy is actually the superset: the basic mode of reasoning and logic that other fields implement to create their more exacting and specific rule-sets. In this sense, mathematics is the philosophy of numbers, physics is the philosophy of massy objects, political theory is the philosophy of governmental structures, etc. This is the original justification behind giving the most experienced practitioners in a field the title 'Doctor of Philosophy' (PhD); the idea is that such people are in a position to wrestle with the fundamental definitions of their particular field, beyond rote application of the principles.
Systematic rigor, epistemic foundationalism, or whatever other term you might want to use here are only 'higher standards' in the sense that they have closed off a particular paradigm as useful for a particular purpose. They are a toolbox designed for a specific set of acts, but like any toolbox they restrict the person carrying them to performing those acts. A carpenter's toolbox is not much use in an auto shop; a physicist's toolbox is not much use in political science. And if we want to make such a toolbox useful in a mismatched arena, we have to step back and reconceptualize the nature of all the tools in the box to make them applicable.
That's philosophy.
Bertrand Russell and the Logical Positivists tried very hard to 'mathematize' philosophy — to give it exactly the kind of rigor and epistemological foundations that you are talking about — and ultimately failed, for all their accomplishments. The mathematics toolbox did not suffice to create a general framework for philosophical questions.
I'm reminded by an inaugural lecture by a Professor of Physics (A. B. Pippard) who posed the question "what is physics", and ended up defining it as "that subset of science which is completely understood", pointing out that when parts of chemistry or astronomy become sufficiently well understood, they get reclassified as physics. Arguably the subset of physics that is completely indisputable similarly gets reclassified as mathematics.
Mathematics is only useful for arguing about the real world when you can reduce the real world to formal abstractions that behave in a predictable and logical way. Philosophy is (often) concerned with discussing what those abstractions should be and whether they are valid.
And of course large parts of philosophy are concerned with questions such as justice, ethics, and politics, where (a) reasoning about cause and effect is never going to be 100% provable (because you can't do the experiments), and (b) half the discussion is about what we want to achieve, not how to achieve it.
Traditionally, mathematics was far less formalized than now, and even when rigor was shockingly lacking (like in the beginning of calculus) and mathematicians didn’t yet possess crisp concepts it still produced insanely much more consensus and progress to virtually undeniable truths than philosophy. I guess it has something to do with the fact that mathematical concepts are way more interconnected and can be looked upon / approached from a multitude of different perspectives. So you simply do not get that far in mathematics with bad reasoning than in philosophy.
As a last resort, mathematics is also an applied discipline: Even if math and stats professors have been tripped up by the Monty Hall problem – you can try it out! This is not possible in philosophy. There is not even a clear standard what it might mean when the application of a philosophical theory fails. Sure, “Marxism failed” is a pretty good argument for many people, but it lacks the decisiveness of failure in applied mathematics … when planes crash.
Well it depends a lot on what kind of philosophy you're doing. The most important split in this case is that between Analytic Philosophy and Continental Philosophy.
Analytics generally are much more concerned with standards of proof, rigorous argumentation, logic, and the like. Think Bertrand Russell. For this type of philosophy, you generally actually do have to make arguments logically formalized and completely watertight.
However, this approach, while it is precise and powerful for certain kinds of questions, cannot answer everything. This is why Continental exists. Let's say I want to answer, for example, "What does the logical desire betoken?" Well I am in fact problematizing the very feeling of need to achieve truth through logical argument, so if I required myself to use logical argument to answer it, I would be operating within that sphere I wished to question. This would be a mistake.
Finally, I wish to appeal to the obvious fact that you can know things---and even, that you can partake of Truth---without logical proof. And many true things cannot be proved (ironically, this last statement actually can be logically proved; it's called Godel's Incompleteness Theorem). Therefore, truth and knowledge does not always require logic.
As well as concurring with @Carduus's answer, let's just look at this one statement:
The same general principles that apply to thinking about the abstract objective universe should also apply to the concrete objective universe.
Even in Physics, one does not prove things about the world. You create models, prove consequences within those models, and then use that to provide predictions about the world. (Questions on the need for these to be falsifiable or not are outside the scope of this thread.)
It wouldn't be philosophy if it uses math. It would be math (or applied math).
What is Philosophy?
Go to any philosophy website (like this one, or SEP), or read philosophy text books or notes -there is one thing which is strikingly common. The usage of natural language.
There are things which aren't developed enough, or not clear enough to allow us to use exact frameworks. We therefore have no choice but to use vague frameworks like natural language to discuss them. This vagueness allows us to discuss. This act of discussion in natural language is philosophy.
If you use math, there are rules, and once you have agreed upon rules, you know the result -what's there to talk about? You don't agree with my mathematical result? Fair enough. Now you want to use some other exact framework like mathematics -go ahead and you are back to where you started. Moreover, mathematics is a logical treatment on complex things. It doesn't tells you what is right. It just tells you that if you believe in your axioms, then following things hold true -on account of being logical consequences. We require natural language for debating, discussing and resolving issues which otherwise cannot be resolved. That's what philosophy is about.
When you are not stating facts or objectively describing the state of affairs, you are technically engaging in philosophy.
Also, read Wittgenstein's works.
Mathematical logic has axioms and/or assumptions as the foundation. I am not clear what you mean by SOUND ARGUMENT. If you as a math person require sense verifiable objects and awareness of truth value, you have made a nonsensical statement. Sense verification is needed for Mathematical truth but you want SOUND ARGUMENTS?
An issue is existential import and objective knowledge. How do we know objectively there are no unicorns? How does one disprove the proposition? All a sense verifiable person can do is require a single instance for a proposition to be true. We don't always have that luxury in reality.
Now I come to YOUR ASSUMPTIONS or AXIOMS in Mathematics. You are NOT saying x is a True proposition outright. You are being slick by placing it in a conditional form (if . . . Then . . . ). This way you can't be said to be WRONG or mistaken as it was just an assumption anyway. Philosophers often have a backbone. If I make a proposition I have to own it if I end up being wrong. If I say all swans are white and someone finds black swans in Australia my position is busted. Too many busted attempts lower my standings in a social circle whether you want to omit this fact folks. ( If I give you wrong directions two times most practical people will not ask me for directions a third time to see if I get it right.) Back to my point: saying x is a true proposition is CLEARLY not as psychological as If x is true then something else is true. You save yourself from embarrassment because you never committed to any REAL CLAIM in REALITY and said nothing basically. If I am wrong with a proposition, then judgement of me and the claim follows. When you blunder you cover it with that huge IF.
Help me understand what you mean by SOUND ARGUMENT when the fact is that many people fail to have a back bone to make a CLAIM about reality? For instance, my proposition IF there were unicorns in New York, they would all be white animals DOES NOT make a claim about reality. How is this format (IF . . .THEN . . .) you run to so reliable and SOUND? We know sometimes this works and other times it fails. If it fails even 1 percent it is not absolute or certain. Science has the same issue. So math taking the scientific approach may be a mistake. You both have the same issue. Senses can be wrong.
I am also a bit confused about the terms ASSUMPTION and Axiom, which where I am from, DOES NOT designate you have any awareness of the ACTUAL TRUTH VALUE. Science can demonstrate all dogs are animals. I do not need an IF Statement to cover myself like you would. I would just have a back bone and say out loud All dogs are animals & accept the backlash or correction when it arrives. You seem to want to avoid all of this action. Pretty slick to me. I find it a bit odd to say Mathematical logic is sound as you say because you can't even tell me upfront if the assumption is TRUE. I guess then it would not be an assumption huh? MAKE A CLAIM not a way out if you are wrong. For an argument to be SOUND the premises MUST BE TRUE -- not kind of iffy, kind of true, possibly true, I am not sure if it's TRUE, etc. Its ironic you want sound arguments but start with a claim you are not certain of. Make a claim first! Tell me the proposition you begin with IS TRUE absolutely. That is all I ask and then we can work on building a SOUND ARGUMENT.
Your understanding of the term proposition is not on par with old school philosophy. All propositions are expressions that have a TRUTH VALUE. Your problem as well as the scientific problem is without sense verification you take the liberty to say a proposition has a false TRUTH VALUE or better yet the UNKNOWN VALUE. Let me get this correct, because you don't know it or have sense verification the value is NOW unknown? The proposition Jimmy Hofa is buried at Shea Stadium under home plate is FALSE because there is no evidence right NOW? I capitalized NOW because time can change as well as technology. Suppose tomorrow Hofa's body was found. What was scientifically FALSE or UNKNOWN magically has another TRUTH VALUE. Is science and math magic? A more rational approach would be the proposition about Hofa being buried was OBJECTIVELY true or false from the beginning --regardless of time or technology. Because YOU WERE UNAWARE of the truth value does not dismiss the actual value nor does new information CREATE A truth value from nothing.
I am not sure what you mean by Sound argument when there is little accountability when there is an error and no necessary commitment to say something about reality. Axioms by definition often fail the sense verification requirements that WAS used about unicorn propositions. I guess we will let that slide and say its objective knowledge. Can other types of reasoning outside of math be useful yes. Is mathematics the most useful? No. The starting point is the problem-- as is the argument of GOD. I can prove almost anything if we always take the assumption as truth. We would have to question what is truth and define truth I reckon at some point. We can all agree bad assumptions will make poor reasoning about reality. What do we do when we assume the negative? Often times we lose our trail and the case goes cold. This further would put you in a place where you dont want to put yourself out in the wind for judgment, but what do I know. I know nothing: as I can't prove most things I claim to KNOW sufficiently and I can never be wrong if I never make a statement about reality; and knowledge is demonstrated by expressing some kind of claim. Expressing something I am uncertain about does not seem a method that gives me 100 percent accuracy.
Any axiomatic system has serious limitations , see Godel incompleteness theorem and related results. The vagueness of language, on which philosophy is based, is its strength, not its weakness. This is the only way to circumvent the serious limitations imposed by any rigid axiomatic system (which can be modelled by a Turing machine). The notions of mistake/error are underestimated , yes, they can lead you astray, but they can also lead you to the truth (with a nonzero probability). On the other hand, an axiomatic system will never lead you to the truth if it's not already "contained" (as a figure of speech ) in its axioms. Of course, that doesn't mean that perpetual nonsense is recommended, it's a matter of moderation.
