Usefulness is the measure, the touchstone of mathematics, and not its logical, conceptual and sensible status - these are merely incidental, tacked on as it were to the outcome we are seeking. The logical and sensible status of mathematics act as ‘sequences of reminders’ for useful processes and for the furtherance of expected outcomes.
Therefore, there are lots of ways in which mathematics finds itself lacking in general validity, but these are never allowed to impinge on the essential criterion of its usefulness. Its lack is found in conceptual matters. One is mathematics arbitrary conception that in a closed system identical things are countable. Another is its confusion of point and position. Another is its haphazard borrowing of metaphysical terms and concepts, such as infinity. These are of no consequence to the essential operation of mathematics, except to act as invocational reminders and pointers. But if by ‘general validity’, we mean whether mathematics is useful, then mathematics is by definition, simply, always useful.

I understand what you’re saying quite a bit better now that you’ve explained and id be almost willing to say its worth discussing.

However, i honestly lack the mathematical expertise. I would hazard a guess however, that advanced mathematics is the BEST way of dealing with such concepts as infinity and such. Its not metaphysical at all, although physics doesnt like it, infinity does play a large role. If you dispute their conception of infinity, you dispute current knowledge of physics. Can you justify this?

As for the rest, i have no idea how philosophy and mathematics deals with them differently.

I would also question your division between usefulness and general validity. Surely for something to be useful (and you really cant underestimate that usefulness) it must be applicable to the real world, so i fail to see how the real world can be manipulated and explained by invalid tools. Doesnt it suggest that the world is generally invalid?

Math is not meant to be connected to “reality” directly. It is a set of rules, axioms and logical processes that can be used in many cases to explain and model the real world. It was never meant to, nor does it have to, have a direct physical analog. In other words, so far, I agree with you…

Now you’ve decided to bring back all the concepts that have been beaten down in other posts of yours…so I have to do it again, for your own good, until you learn.

When would identical things not be countable? you must have at least 2 things to even decide if they are the same…so one of them magically dissapears when you realize they identical? Explain this…

As far as I know, you are the only person who confuses point and position. Math draws no distinction because THERE IS NONE. A point is the 0-dimensional “object”, whose position, is described by a set of coordinates. Ordinarily, this implies a distinction between the two. However contains no other property than it’s position. It is defined only and entirely by it’s position, therefore, a point and position are equivalent.

The only confusion is when people use “position” to incorrectly describe an 1+ dimensional object. Obviously, an object with a finite size has no single position…whatever line, area or volume it encloses describes a range of positions and points. We might say the center of this object is at x posioont. the center is a point, so that is valid description.

Mathematics rarely deals directly with “infinity”. Calculus, for instance, mostly describes what happens as you approach infinity. This has direct applications to the real world; acceleration is the derivative of velocity, which is the derivative of position.

Just because infinity does not occur in the real world (on earth i mean), that doesn’t automatically make it invalid. Heck, by that logic, I could say metaphysics is totally invalid for the same reason. All infinity really means is “not finite” , in other words, not bounded. We can understand this…look out into space for an example. Obviously, since it doesn’t have a boundary, you can never “reach” infinity. you are always approaching it. Limits, and everything based on limits, uses exactly that concept.

I spent one year studying the philosophy of art. It was not a demand of that course that one should know anything about art, above and beyond common familiarity with it. Similarly with mathematics. The deeper you go into the philosophy, the less you need to know about particular cases. There are general forms that mathematics uses that mathematics never talks about. For example, even the simplest equations use mapping, referencing, the setting of arbitrary and extraneous limits, and identification, though you won’t read about it in a mathematics text book. Mathematics will talk about limits, and succession of number, but misuses these applications when it talks of largest numbers, infinity. I am trying to get hold of a copy of Frege’s Foundation of Arithmatic. I expect this not to be a mathematical study but a philosophical study of mathematics. There is a difference.

My point was that mathematics is defined by its usefulness. The concepts it uses, the shapes and pattern recognition of equations, the metaphysical invocations used in number theory (and quantum theory) are quite inessential to the operation of mathematics which is geared to outcome.

One possible example where the metaphysics of mathematics misleads is in quantum theory, where the interpretation of the mathematical outcome, with an exaggerated emphasis on its metaphysics, leads to the ‘probable worlds’ of quantum theory.

Quantum Mechanics is not misleading. Scanning tunneling electron microscopes are tangible proof that quantum mechanics correctly describes particle behavior, at least more correctly than anything else we’ve come up with. The particular effect is called barrier tunneling. An electron trapped within supposedly impassible boundaries has a small but finite probability of appearing outside of the boundaries.

In the case of the STM, the Boundary is the space between a small scanning tip and the surface of the object being scanned. As the tip passes over individual atoms, the thickness of this space boundary changes, which in turn varies the rate of barrier tunneling. This rate is recorded wrt position of the tip, giving a “picture” of individual atoms.

Quantum mechanics is technically a model of the particles’ behavior, not necessarily the actual mechanisms bywhich particles interact. However, in practice, having a good enough model is an acceptable substitute for knowing exactly happens, and so quantum mechanics is considered to be the way things work until something more accurate comes along.

tmminionman2.
Minor point: space does have a boundary. The universe is not a bunch of stuff expanding in a void which is space. Instead, it is a bunch of stuff in which the space between each point is growing. Consider the surface of a balloon as you inflate it, the stuff inside is space, the stuff outside doesnt exist and the membrance is the stuff.

[ignore] I also question the statement that mathematical principles are not directly applicable to the physical world. Again, as quantum mechanics suggests, the more you look at the universe the more it is about the exchange of information and less about the interaction of ‘things’. In fact ‘thingness’ itself is just a property of that information. Maths is an abstract yes, but so is quantum mechanics as opposed to common experience. [/ignore] <EDIT: i didnt see your next post, and must have misunderstood your original statement>

JJ.
Last Paragraph: you bring into doubt the scientific method too? Of course quantum theory is extraordinarily philosophical but i wouldnt say its based on any traditional metaphysics. Though im ignorant of your example, i gather stuff like “probable worlds” is just scientists playing with philosophy rather ineptly and i doubt it is the concensus.

PS. I promise never to bring up quantum mechanics to support a philosophical statement ever again… for the next two days… or less.

The idea that a world is probable, and that it therefore exists in a universe populated by possibilities… if this idea is shorn of its jargon, all we have is a hidden, unobserved world from the one we live in. Just like Valhalla. And the void of Steven Hawking? Gunningagap, the realm of the densely packed, and other mystical teachings of many cultures have posed this. All that is different is that these ‘places’ are populated, reached, or emerged from, by different objects and entities.