Medieval scholasticism? :roll:

Is Arche more akin to "first causes" or axioms or postulates from which first causes might emanate?

the Department had to attract more students, and so was to both accept students with less ability and offer less demanding courses — Banno

We (mathematics) were mostly a service department, with courses we designed for liberal art majors, business majors, engineering tech majors, etc. We occasionally had run-ins with those departments about grades and standards, and we could adjust those without abandoning our self-respect. I mostly became responsibly involved in these things when I headed up the department for a couple of years.

Because of the service factor we were able to keep courses for our majors at a reasonably high level, with me designing and teaching the senior level offerings in real analysis (intro), complex variables, and topology. I was considered the most demanding, but mostly about giving a B instead of an A. :cool:

The result is apparent in this forum. Folk think philosophy easy, a topic for dabbling dilettanti — Banno

I plead guilty. I am learning slowly about what makes a philosopher tick. My one senior level course in the subject in 1958 was more of a survey and little was said about the practice of philosophy.

It's just preferable to argue about the meaning of "gavagai" on a full belly. That's pretty much the reason i decided not to pursue academia — Banno

You must have had a feeling of knowing what was in store. Or was it more than that? :smile:

Seems to me that there is a play on two senses of "know" going on here — Banno

And the distinction between "knowing" and the feeling of knowing. When I follow the proof of a theorem I know the theorem is true according to the rules of the game, and I have a feeling of knowing. But when I park my car out of sight I only feel that I know where it is. So it is conditional knowledge. Most of life is lived in a complex of probabilities.

But then I have the feeling of knowing what I have written is of no consequence in this forum. :roll:

Interesting link. It's not quite what I had in mind, but probably more reasonable. I had thought of a continuum between the world of ideas and the physical world.



Thought-provoking. Thanks. Goes back to constituting mathematical (and physical) reality from the continuum to the discrete, not the other way around. Traces of Bergson.

Mathematics, it seems to me, is like them in that respect - it adds to its traditions without superseding them — Ludwig V

That's the way it usually works. Math people "create" or "discover", building more or less on preceding results, sometimes obviously but at other times seemingly "new".

It is true, of course, that mathematics often turns out to be useful, but I can't accept that that is its point — Ludwig V

Bingo! :clap:

Top Ten TV series??? — 180 Proof

I'm so old I forget what I've watched. But some are Rake, Lillehammer, Luther, White Lotus, Veronica, True Detective, Fargo, Money Heist, You, Sopranos, and on and on. I enjoyed them all.

I haven't been to a movie theater in at least ten years. With a big screen TV and HBO, Showtime, etc. plus Netflix there is an overabundance of entertainment. I prefer series over films for the opportunity to develop personalities and plot intricacies.

AMC theaters will soon charge more for prime seating. Maybe they'll bring back ushers with small flashlights that theaters had sixty years ago.

No. I meant to say "with depth" — L'éléphant

And you did. My mistake.

I really do not believe that thoughts are even similar to material objects which I also call "things". With talk like this, we create an environment where ambiguity and equivocation are highly probable — Metaphysician Undercover

Yes, and in the quantum world those distinctions could be imperiled. The problem of actualization of potentia brings science and philosophy forcefully together IMO.

Do you really need me to explain to you what I said in english? There are things you could say with depth about the subject besides "Over two millennia have passed with no consensus". — L'éléphant

Maybe in English. And I'm sure you are correct. I suspect most of those things "said in depth" have relevance in philosophical circles rather than in mathematics communities - or anywhere else. It's good to know the limitations of one's reach. I have created and proven perhaps two hundred theorems - but they are virtually worthless, lost in millions more. All said in depth. :cool:

The study of mathematics is not the same as the study of philosophy — L'éléphant

Consensus is vital to mathematics, but from what you say a hindrance to philosophy. When one argues about the reality of numbers, that is not an argument in the realm of mathematical practice. It may have great meaning for philosophers but is seen as incidental to the subject by most math professionals. On the other hand, a philosopher might have difficulty explaining philosophical implications of a theorem picked at random.

There's no debate, Wayfarer. Mathematical objects are quite real for me, though not like Lake Michigan is real. But I do think it's possible that the mysteries of where math "converts" to physical reality (well, I'm grasping at straws here) in quantum theory may shed light on the subject far beyond what philosophers and mathematicians have thought to this time. That's where I see "the profound issue". :smile:

Over two millennia have passed with no consensus. — jgill

Jesus. No disrespect, but if this is all you could say about philosophy, then you don't fit in philosophy — L'éléphant

So, you are saying there has been consensus about the reality of numbers and whether math is created or discovered? I'm not addressing other aspects of Platonic philosophy.

That is what I would describe as a jaundiced view — Wayfarer

I agree. It is. On rare occasions in my career when the nature of the reality of numbers and math came up amongst a group of my colleagues invariably eyes would roll and the topic would disintegrate shortly thereafter. Had I been among foundationalists reactions might have been different.

Look to the future, not the past, pave the roads for all to achieve what they are capable of. I never complained about the idea of affirmative action, only in its interpretations and practice at times.

Work on philosophy -- like work in architecture in many respects -- is really more work on oneself. On one's own interpretation. On how one sees things. (And what one expects of them.) (Culture and Value)

OK, I'm on board with that. :up: :smile:

Philosophy is concerned with what was said or printed or argued in the past — jgill

Not in my view, obviously, but I won't try and persuade you — Wayfarer

Isn't that what you are talking about? The issue of the "reality" of mathematical objects. Over two millennia have passed with no consensus. When we speak of Platonism isn't that something from ancient times?

However, quantum theory may ultimately bring some clarity as physicists explore the mysteries between mathematical entities and physical reality. Where in a process does actualization occur? Virtual particles appear to be Platonic rather than physically real - they cannot be observed and yet they are convenient in certain procedures. In QT is where I might expect to see progress in understanding the nature of mathematics, here is where the subject may morph into a kind of neophysical existence. Who knows?

How large a coin would it take for you to squat down and pluck it from the ground?

A nickle for me, a penny for my wife.

Thanks. My question was about the sense in which a domain, such as the domain of natural numbers, is real, but not phenomenally existent. I notice that nowadays it is commonplace to say of anything considered real that it must be 'out there somewhere' - but even though such a domain is not anywhere, it is nevertheless real. See this passage. — Wayfarer

From your link:

Cunningham had unwittingly re-ignited a very ancient and unresolved debate in the philosophy of science. What, exactly, is math? Is it invented, or discovered? And are the things that mathematicians work with—numbers, algebraic equations, geometry, theorems and so on—real?

It may be ancient and unsolved, but that doesn't mean it holds the interests of those involved. What appeals to most mathematicians is the exploration and creation (or discovery) of new ideas - new theory. I was a rock climber for over half a century and what compelled me in both math and climbing was finding out what lies around the bend or over the overhang, whether it's creating or discovery - an argument that few in the profession care diddly about - and this seems to be a major difference between what is being said about philosophy in this thread and what is true of mathematics, save for those few in math foundations: Philosophy is concerned with what was said or printed or argued in the past, whereas mathematics (with the exceptions of a few math historians) always looks toward the future, even when analyzing the present lay of the mathematical landscape.

From "what is new is the exception" to "what is new is the rule".

Foundations and set theory, overlapping philosophy and mathematics, are out of my bailiwick. :cool:

Thus, the physical theory of dynamical systems could be transformed into a model in which that consciousness could be described as an attractor, the physical concept of information (not Shannon!) in connection with information or structure density describes the same dynamic 'center', . . . — Wolfgang

Such a model would be interesting. I've played with dynamical systems in the complex plane for years, and have seen only one instance when a theorem I contributed was extended to the realm of psychology (decision making in groups) - and this was done without regard for the technical hypotheses. That's what seems to happen when math is appropriated by a social science. I doubt things would be much better in a quasi-biological setting.

Your replies are always entertaining, and frequently thought provoking. Continuous mappings, in the context of QM, can mean the topological definition applied to the Hilbert spaces of that subject. Inner products yield norms which give rise to metrics, within which continuity is defined. It's a long way from Aristotle.

I still think much of what is discussed in this forum concerning QM boils down to the problem of unitarity.

I think the issue with the lattice representation is that the designation of a quantum (discrete unit) of space is completely arbitrary, not based on any real attributes of space itself — Metaphysician Undercover

What are the "real" attributes of space?

. . . and internally as will. — KantDane21

Interesting. In my experiences in the Art of Dreaming (lucid dreaming) I become pure will.

I have even heard it said that in philosophy, getting it right is less important than being wrong in interesting ways. — Ludwig V

Delightful! Thanks for a glimpse into the profession. :cool:

then the physicist will continue into that theoretical fantasy land, a fictional world requiring the assumption of "virtual particles", in a pointless attempt to maintain the representation of mass at a point. — Metaphysician Undercover

Lattice field theory avoids virtual particles, which are mathematical conveniences.

the point does not provide a very truthful, or even accurate representation of a body, which really exists in the area around the point — Metaphysician Undercover

I'm curious how you would express what you have said in the context of field theory.

The question is not one about physics, it's one about meaning. — Wayfarer

Unfortunately, it may well be the meaning of the math as one follows that path to actualizations. When does that occur? Beyond me. The math is hard for me to follow. Not so young, anymore. :worry:

Every single moment we're reminded of how small we are while our hearts & minds yearn for the great. Soul-crushing it is (for those who recognize the problem) — Agent Smith

Nailed it, Dude. :cry:

:lol:

The use of "inner" makes it sound like these are properties internal to the point. In reality they are how the point relates to other points (by means of vectors), therefore external relations. — Metaphysician Undercover

Inner product = input vectors into a form producing a real or complex number (scalar). Outer products are matrices or tensors. Word salad.

The problem with vectors is that they represent things (forces and movements) with one dimensional straight lines, when we know that in reality these things act in a multidimensional way. — Metaphysician Undercover

The issue is, as I said at the beginning, the straight line of a vector does not accurately represent a multidimensional activity which has curves inherent within every infinitesimal point. So real movement from one infinitesimal space to the next is not accurately represented with straight vectors, and the longer the vectors are, the more the inaccuracy is magnified. — Metaphysician Undercover

Oh my. This is dreadful, I fear. :gasp:

Whereas the simplest vector spaces (in R^2 or C) have vectors which can be represented by little arrows in the Euclidean or complex planes, most vectors in QM go far beyond this and cannot be so described. See Hilbert space. But, if I read between the lines you write I think what you may be getting at is the fact that linear maps are fundamental in applications.

Working with complicated functions in math one frequently tries to approximate little parts of these functions with linear functions, which are so much easier to work with. That's what happens, say, in finding a distance an object has traveled, D=Rt. If R is constant, we have R(t1+t2)=Rt1+Rt2. But if R varies we go to a definite integral, which, itself, consists of adding tiny parts of time with constant rates applied.

I've wondered about this linearity feature of QM and why it is so fundamental to the subject. But I am not a physicist. Here is a comment found in Wikipedia.

Oh Brother, Where Art Thou? is my favorite of all time. I've watched it a dozen times. Mostly the wonderful music. :smile:

Politics is easier than science. The reason: It's easier to lie and get away with it in politics.

A physicist, especially one who is a genius, can manage to be an eccentric, offputting, raw truth blurter with poor hygiene, even — Bylaw

Further a physicist might be terrible at reading people's emotions. They might react with tremendous confusing when encountering subcultures other than their own and might have no interest in trying to understand them — Bylaw

They might be utterly incapable of speaking in different ways to children, poor people, working class people, rich people, people going through trauma and so on. Another way to more neutrally put all this is they could be socially rigid. You could say, such a physicist is socially honest. Or you could say they are a very poor communicator. — Bylaw

Whew. You have a thing about physicists. The ones I've known had none of these characteristics. :roll:

Philosophy is self-reflexive and dialogic. What others have said is not separate from what one says about world, existence, reality and truth.

Original ideas and concepts have always been the exception. — Fooloso4

Thank you for this moment of clarity. In math what has come before is a stepping stone to advancements. The actual words of the pioneers are immaterial.

The word domain has several meanings in math. In number theory you have integral domains and when speaking of functions of one sort of another it refers to a set, let's say, of x's for which f(x) is defined.

Basically, it's just a set of objects defined in some sort of space. Not meaningful by itself.

Have you ever happened across Wigner's essay — Wayfarer

Yes, some time ago, but nothing stuck, so I guess I wasn't moved by it.

↪jgill
Coming to think of it, here's a legitimate question within your area of expertise: there is a 'domain of natural numbers', is there not? And there are numbers outside that domain, like the imaginary number −−−√1 which is used in renormalisation procedures in physics. — Wayfarer

You must be referring to integral domains, commutative rings, generalized from the properties of natural numbers. From natural numbers one enlarges to rational numbers, then real numbers. And then extending this to complex numbers. Complex numbers are used as a powerful tool in QM.

Is there a question here?

I think that deep within the mathematical structure of QM is where superposition or assumed existence in two "separate" states simultaneously occurs, or a mixture of states. The process begins with Hilbert spaces and their inner products (in C these might be thought of as the power of combined vectors). States of a system are subspaces of these. A pure state is determined by a unit vector in the Hilbert space. Combining systems is interpreted as taking tensor products of two Hilbert spaces. The probability of a property when the system is in a pure state is given by the inner product.

It seems to be a mathematical thing and perhaps someday a different math approach will clarify this. Schrödinger's cat deserves a bowl of milk and gentle scratching around the ears.

Thus we have agnostic realism about the mathematical world: numbers are real but we must be agnostic about the intrinsic character of numbers—as we must be agnostic about the true nature of what we call “matter”.

Well, I would say the "intrinsic character of numbers" is irrelevant to the subject, and an unfruitful environment for agnosticism. Can philosophy bring any clarity to something that exists only within its practice?

I posted a question about philosophy of maths and the ontological status of number, which was frozen because, the moderator said, there was no-one there qualified to address it — Wayfarer

I wonder what would happen if this were posted on a math forum. Probably the same result. There do seem to be ontological questions arising about sets, and numbers can be interpreted as sets.

I am somewhat saddened that the logic and philosophy of mathematics and philosophy of science categories never receive much attention or forum posts. — Shawn

The philosophy of mathematics is largely foundation theory, and this is a very technical subject. I was a math prof but beyond naive set theory I know little of foundations. In the past the forum had several participants who seemed quite knowledgeable in the subject, but, apart from Tones in a Deep Freeze they don't seem to be active. Beyond foundations I suppose one looks into the historical origins of the subject, arguing what Aristotle really meant by something attributed to him, etc. Not much there in my opinion.

As for science, threads on quantum theory spur a number of posts, many of which appear authoritative, but I have my suspicions. We have had a few actual physicists active here, but they seem to have at least momentarily fled the environment. It's an arena of discussion that beckons those who enjoy batting around the quaisi-woo some actual luminaries lay out - seriously or frivolously.

Philosophy is laugh-out-loud good times for those who love it, especially in the heat of battle with all marbles on the table. — ucarr

I've wondered what happens to those when a philosopher loses them. Now I see where they end up.