And thus on that day the Lord created aman — Kenosha Kid

Whoa, not so fast! Where is the link between complex numbers and the existence of man?

Inquiring minds want to know. :chin:

A matter of notation and mathematical clarity. See definition 7.1

https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Lian.pdf

Not a big deal. You are to be applauded for wading into this.

x=(a,b) and y=(c,d) are ordered pairs of integers, not integers themselves. So x and y belong to the Cartesian product of the integers. You're defining an inequality of ordered pairs in terms of an inequality of integers.

Calling fishfry! :cool:

Christ, I think it'll take me longer to debug the mathjax than it did to write the comment. — Kenosha Kid

MathType works well - it's WYSIWYG - using the Wikipedia cut and paste option. You only have to replace <math> with [....]. Just a thought.

∀x=(a,b),y=(c,d)∈Z
y>x↔c+b>a+d — Kenosha Kid

$\left( c,d \right)\in Z'=N\times N$

Trump's male picks all seem to be spineless, vicious, or both — tim wood

I'm curious about Neil Gorsuch in this regard. Is he really that bad?

The point of that thread is to illustrate specifically how, in a very distant way, we ourselves can be said to be made of empty sets. — Pfhorrest

Some time back on another thread I mentioned being curious about the process of going from the empty set to the fundamental theorem of calculus, step by step. I had done this sort of thing sixty years ago when a professor had us reach the exponential function this way. Your proposal is much more challenging. :smile:

Here the case of climate change is especially relevant. This is not an issue about abortion, guns, or immigration -- this is an issue that is settled — Xtrix

True enough. What is not settled is whether it can be altered significantly. Especially with not all nations on board. Don't buy beachfront property in Miami.

This is a very ambitious thread. But probably no more than the first 900 pages of Penrose's The Road to Reality. I may not live long enough to see its completion, but you guys are younger, so there is hope. :worry:

I shall try and get the field of real numbers with its order defined in that time. — fdrake

Look forward to this! Next, well-order an uncountable set of reals. :cool: :up:

I'm picking up tiny tidbits of math here that relate to topics I am familiar with, but it's all pretty fuzzy. For instance, $SU(2)\subset GL(2,\mathbb{C})$, and the matrix multiplication in SU(2) corresponds to compositions of bilinear transforms - as I mentioned before. To elaborate:

$\left [\begin{matrix}{{\alpha }_{k}} & -{{{\bar{\beta }}}_{k}} \\{{\beta }_{k}} & {{{\bar{\alpha }}}_{k}} \\\end{matrix} \right]$

Corresponds to

${{t}_{k}}(z)=\frac{{{\alpha }_{k}}z-{{{\bar{\beta }}}_{k}}}{{{\beta }_{k}}z+{{{\bar{\alpha }}}_{k}}}$

And multiplication is the following

${{t}_{k}}({{t}_{k+1}}(z))=\frac{{{\alpha }_{k}}{{t}_{k+1}}-{{{\bar{\beta }}}_{k}}}{{{\beta }_{k}}{{t}_{k+1}}+{{{\bar{\alpha }}}_{k}}}$

edit: But the matrix represents a quaternion, primarily. I.e., extending the complex numbers to a higher dimension. In the above, alpha=a+bi and beta=-c+di to give the quaternion a+bi+cj+dk.
SU(2) seems to associate with spin. The multiplication of two such 2X2 matrices may give the Hamiltonian product, or maybe not. Too much work involved.

GL(2,C) is a Lie group corresponding to the group of linear fractional transformations (LFTs). When you matrix multiply two 2X2 such creatures it's the same as composing one transformation with the other. Each such transformation maps Circles to Circles in C (a line is construed to be an infinite circle), so these transformations preserve those geometric objects in the complex plane. Otherwise, in the complex plane simple translations and rotations preserve geometric figures. I've done lots of exploring of the properties of LFTs, but all of it as analytic or meromorphic functions from an analytic rather than an algebraic perspective. Very old fashioned
of me.

@fdrake is far more up to date and knowledgeable.

Better simply to eviscerate the "theory". :yikes:

Different coordinate systems can map different numbers to different points without changing any features of the resulting geometric object — Pfhorrest

You probably need to qualify this. Take the circle x^2+y^2=1 in the standard Euclidean plane and lengthen the scale on the x-axis, so that the circle becomes an ellipse. That's a "different coordinate system".

Amazing how the physicists can use those groups. :cool:

Maybe Philosophy will always only describe the structure of reality without actually participating — Noble Dust

I see occasionally on this forum discussions that might go on for pages about subjects that are actually practices and not merely ideas. When that is the case first hand knowledge seems to me to be a prerequisite for philosophizing. But I may be in the minority here.

↪jgill
The Lounge isn't supposed to be purely philosophical. You can start a thread solely about kittens here, according to the description. — Professor Death

I think I made that post when the thread was on the main page. More appropriate here.

And it's not a matter for philosophical debate — tim wood

Then it should appear in a different forum.

So the speculative metaphysics I had in mind is a physics where spin is the basic notion — apokrisis

Thanks for the clarification.

I'm a retired mathematician with little knowledge of philosophy, so I mostly read posts. The writing on this forum is at times excellent, and the ideas expressed - well, sometimes consistent with the quality of writing. Oddly enough I have learned a bit more about mathematics here, as well as physics. But I admit I get snookered by convincing posts in the latter subject when metaphysics seeps into the discussion. :smile:

I am basically asking if we gain any real virtues with age that aren't the direct result of our own decline? — TiredThinker

In my 8+ decades I have seen and experienced much, and that accumulated history perhaps gives me a different if not better perspective on current events. I have become more tolerant and far less impulsive.

And since Frank Apisa was excommunicated from the forum I feel somewhat alone. That's not bad really, since much of my avocational career as a rock climber was going solo. :cool:

Take the law of universal gravitation: G(m1*m2)/(r^2) ... For instance, it could've been G(m1/m2)*r^3 — TheMadFool

How intriguing . . . please explain.

Probably. But take for example the element of the power set of N: {2,6,7}. This could be interpreted as guest(2)->room(6), guest(6)->room(7),guest(7)->room(2). Kind of silly, I guess.

Wiki, Sartre's The Look: "The mere possible presence of another person causes one to look at oneself as an object and see one's world as it appears to the other. This is not done from a specific location outside oneself, but is non-positional. This is a recognition of the subjectivity in others."

This is deeper than simply not caring what others think. As a former outdoors athlete I experienced this many times. And it wasn't related to safety or competition issues. The mere presence of another changes the experience.

A highly colored painting turned to pure white leaves nothing but an empty canvas.

I don't remember ever coming across a claim that there's an operation we can perform on the naturals that can yield a greater infinity than it. — TheMadFool

The power set of the naturals.

One long, continuous paragraph might be a little much to process, whereas ten second parts separated by pauses allowing the listener to digest a sequence of related ideas might work better.

I wonder how the kids of today are going to do when the have to sit through a 90 minute university lecture. — Sir2u

Times are changing. Ninety minute lectures may fade away. I hope so.

Does this attitude/behavior reveal something about the human psyche? — TheMadFool

It might reveal more about what you were saying during that 59 seconds. Just a thought.

On an uncrowded roadway I might turn the driving over to "George", who is a part of me. Sometimes George takes over when I back out of the garage and a few minutes later I wonder if he closed the garage door. Generally, George and I get along fine.

But George is not God.

I've got plenty of nothing — magritte

and nothing's got plenty of me!!!! :party:

How can a sphere be unbounded? — Gregory

Riemann sphere

↪Gregory
But maths treats infinity as a "discrete" whole. You have infinities of many different "sizes". — apokrisis

This certainly seems to be true of areas of math concerned with sets and/or foundations, and probably valid in other areas of modern, abstract mathematics including analysis. Although in my subject of classical complex analysis a specific point on the Riemann sphere identifies with infinity, I've never had to deal with a discrete whole infinity and think of unboundedness in the complex plane instead.

I believe that even when some middle phase is very evident and visible, but it gets traversed, there has to be some explanation as to why it gets traversed — SaugB

It is a victim of the momentum of time itself. Whatever that is! I suppose if time has a kind of mass, then the expression is not completely bonkers. :cool:

The Planck scale is the birth of the dialectical contrast between the reversible and the irreversible as an actualised physical reality. So Bergson was right about durations. Or if we are to talk about point-like "instants", then we have to recognise that they must already have this internal dialectical structure. An instant already marks the point where irreversibility AND reversibility have just entered the world as "a thing". — apokrisis

This certainly gives Planck time a new spin in my thinking. I've assumed it had more mundane characteristics in terms of light traveling a tiny distance and the four universal physical constants. I assume your first sentence above refers to an inability to measure below certain limiting dimensions.

So, how do we attribute existence to that traversed thing, ie, in our example, the orange between the yellow and the red? — SaugB

And how do we attribute existence to the moment in time at which that color exists? Does time flow in a continuum of instants? Or does it exist only in intervals? Bergson argued that time as we live it is in duration (durée réelle), and time for science is a matter of instants - allowing for the freezing of time for purposes of calculations, like instantaneous velocity. So the existence of "a traversed thing" is equivalent to an instant of time.

Peter Lynds had a paper published in the Foundations of Physics Letters some time back in which he proposed there are no instants of time. Some physicists thought his ideas were profound, while most others considered them rubbish. This discussion of time is analogous to dialectics concerning the existence of irrational numbers.

There is new evidence that the so-called laws of physics aren't even constant throughout the universe. You're part of the old school, which is just now beginning to get bumped out. — JerseyFlight

The possibility that certain constants in those laws might vary a bit in space and time does not mean physical principles are endangered. There remains quite a bit of orderliness in nature.

. . . what if symmetry isn't part of the equation, what if we are discovering chaos? — JerseyFlight

We must stay calm. :gasp:

Thanks. Wish I knew more about modern physics.

I took a year of physics in college 65 years ago and of course used some of the elementary ideas in calculus courses I taught. The group SU(2) is as close as I get to contemporary physics as it concerns 2X2 matrices of real or complex numbers. The multiplication of 2X2 matrices corresponds to composing two linear fractional transformations, and I have long explored infinite compositions of LFTs. Apart from that it's Greek to me! :cool:

And if that space is made of, say, unitary matrices instead of ordinary numbers, you can make something like a special unitary group, in which you can build structures identical to the physical stuff that makes up our universe — Pfhorrest

Maybe a reference would help.

I suspect that these were more likely middle-class than working class — unenlightened

In the late 1940s Joe Brown and his companions initiated the British era of working class climbers, and those I met in the 1980s and 90s who were living on the dole were largely, but certainly not entirely, from that class.

If one visits some of the poorer areas of NYC they might find unemployed young men exercising and playing basketball during the working day. Well, before the virus struck! A leisure class, but not an appealing one.

The comment I made initially is not original. It was coined by one of the California climbers of the 1960s - perhaps Eric Beck - about dirtbaggers and elitists of that era. I was a member of that climbing generation and climbed and camped with one of those dirtbaggers from a working class background who is now a billionaire. Another colleague and friend from a humble background became almost literally Royal Robbins: Spirit of the Age.

I've known a few climbers too in N. Wales, and they climbed the slate quarries for fun, precisely because they were not the children of the quarrymen who climbed them with drills and explosives for a living. — unenlightened

Your reasoning here leaves me bewildered. When you are so confident about motivations, you should know something about the world to which you refer.

Leisure: "opportunity afforded by free time to do something."

If one is on welfare, with no job, this might apply. I have observed this in young British men who are addicted to the sport of climbing. They pool their resources for housing and food and pursue their dreams. (However, my observations are dated having been from the 1980s. Thatcher's government might have made this less likely).

Many aspects of the universe are orderly. We invented math to model these features. Why does this orderliness exist? Good question. :chin: