Actually $0^0$ is called indeterminate and has no value. Any rule you're trying to use to assign a value is not applicable.

I have no idea what you're talking about (BS in Electrical Engineering, higher degrees in Mathematics).

How a reflection is constructed :

Given a point of reflection A, let B be any point in the infinite domain containing A (the domain can be a line, plane, or space). Draw line AB and let x be the distance from A to B. Find point B' on AB which is distance x from A but on the opposite side of A from B. Then we say B' is the image of B under a reflection in A.

If you admit that the definition of "opposite" includes "across from", then clearly B' is opposite B with respect to A.

Now consider point A itself. Following the construction given above, A' (the image of A) will be the same point as A (here, distance x=0). That is, the reflection of A in A is A. Thus A is opposite to itself with respect to A.

So, let the domain be the number line and replace A with 0. Clearly each negative number is the image of its corresponding positive value under a reflection in 0 (and vice versa). Now here's the kicker : 0 is a reflection of itself. I.e., 0 is opposite (across from) itself.

By the way, 0 is neither positive nor negative, so let's drop that nonsense now.

Okay, I'll give it a go. But you usually dig your heels in and refuse to hear otherwise when it comes to math. Try to have an open mind.

I could offer an intro to group theory to prove zero is an inverse of itself, but I don't think that's going to sway someone so math-phobic. Let's stick with the idea in my previous post : Can we agree that "opposite" sometimes means "across from"?

To be across from something means to be reflected in a line, point, or plane. Even when facing a friend at a table we can be said to be reflected in an invisible plane between us (actually reflected in a line to preserve left- and right-handedness).

What's of interest is what happens to points lying on the line (or point or plane) of reflection. Under the reflection, such points do not move! Thus a point on the surface of a mirror will reflect onto itself!

When a reflection in zero is performed on a number line, every point maps to it's negated version, but zero maps to itself. In other words, zero is across from (opposite to) itself.

Major Edit : "Opposite" is perfectly fine when discussing positives and negatives. One of the meanings of opposite is "across from". Consider the number line with zero as the value between the positives and negatives. +5 is across from -5. Opposite works.

Thank you. — Metaphysician Undercover

Think nothing of it (much as you do with math).

In the past I have implored you to write up your math musings and send them off to prestigious journals. The math world languishes without benefit of your folk wisdom. Have you ever followed up on that suggestion? Your fame and fortune await.

Hee hee. Welcome to MU, the school of bizarro math, where up is down and black is white. There's only one teacher and he never studied math himself, but he knows what he knows.

Things are going to get pretty bad, and you may get caught up in it. But like cockroaches, a few of us will survive. To fight pessimism, assume you'll be one of the lucky ones. Then our three-eyed, ape-like progeny will inherit a new world.

Did that help?

Um, a bit muddled.

So a mechanism has sensors, and an organism has senses. And an organism's senses are different from sensors - senses do not transmit signals (or they would be sensors). Is that right? Then what do senses do? Do the senses initiate action?

To be kind, your definition of consciousness is ... unusual. As I said before, it puts you in a tiny minority.

A bacteria's reaction to a stimulus seems purely mechanical. The reaction is entirely predictable (within the parameters of the given situation). And no decision-making seems to be taking place. I fail to see how any of its responses to stimuli could not be modeled by a fairly simple machine or computer program.

But then I reread this :

Sensors, on the other hand, are mechanical devices that receive and transmit signals. Like the brain. They don't perceive things.

Oh, now I get it. You think of consciousness as the ghost in the machine.

When you want to claim consciousness for plants and bacteria, I think you place yourself in a tiny minority. To repeat : conditioned behavior does not require self-awareness or high cognitive functions.

Simple machines react to stimuli from their environment. Consider the thermostat.

Yeah, I think there is a gradation in consciousness as well. Had too many intelligent dogs to think otherwise.

Not so sure I'd include plants and bacteria though. Those articles you cite are interesting, but I don't believe very many biology departments are teaching about intelligent plants. At a minimum, I think you need a somewhat sophisticated nervous system. Birds and mammals, sure. But even insects and fish should give pause. Conditioned behavior does not require self-awareness or even other higher cognitive functions.

Fair enough. Truth be told, I believe in a gradation of consciousness as well.

So life is conscious, all else is not. Based on what evidence? Plants? Amoebas? Bacteria? What besides personal prejudice proves consciousness?

Machines require sustenance (fuel). Machines can be designed to move towards light. Etc.

Perhaps. We'd have to meet the monkey.

To avoid solipsism, we all assume that other humans share the trait we call consciousness - and we deny this trait to non-human entities (sorry panpsychists). But, when feet are held to the fire, it's nearly impossible to explain why. Clearly neither appearance nor behavior are adequate to recognize consciousness. (All we ever have is sample of observations. The next time you speak to the entity you call "Mother", her head might spin around and springs and wires pop out from her ear!)

I mean this seriously : What will we do if we someday meet a space-faring race of lizard men? Assume they're not conscious just because they're not human beings?

You're probably right that linear computer systems will never be conscious. But what about massively parallel, mutable computing systems with multiple feedback loops? (I.e., don't go bad-mouthing the internet when online!)

Stevan Harnad points out that Turing wished to change the question from "Can machines think?" to "Can machines do what we (as thinking entities) can do?".

Judging efficiency requires the user to recognize some difference between the two black boxes AND place a value on those differences. Five differences that a user might value that I can think of off the top of my head are size, speed, energy, accuracy, and the expense of creating the boxes. An obvious difference that probably carries no value for the user is external color.

The situation is made more difficult by multiple differences. Is a mainframe that takes up half a room but can solve a difficult problem in seconds more or less efficient than a desktop that requires half an hour to solve the same problem? It depends on the value the user places on each difference (size vs. speed).

And of course judging between a human and a machine is many times more difficult.

The two sides of your equation are, of course, equivalent, but they may not share the same "efficiency" - if you mean by efficiency how much time or energy is required to arrive at an answer. Subtraction may require more steps for a given system (blackbox, human, or other) than addition. Ditto computing sines or cosines.

In the immortal words of the Boss :

Got a wife and kids in Baltimore, jack
I went out for a ride and I never went back

The problem is that you're focused on what happens to the previous occupant of room 1. Whereas your compassion is commendable, that's not how subtraction works. All that matters is the resulting state of the hotel.

Let's say you're holding 3 cookies and I take one from you. Does it matter to you whether I give the cookie to someone else, eat the cookie, or throw it on the floor? All you care about is that you now have 2 cookies.

Similarly, all the hotel cares about is that room 1 is now empty and available. Where the previous occupant went doesn't matter.

It's no difference at all. For the number system (i.e., the hotel and it's occupants), the resulting states are identical (room 1 is empty and every other room is occupied). There can be no hallway - that would be another room. When a person leaves room 1 without entering room 2, they have effectively left the hotel. As far as the hotel is concerned, they have ceased to exist.

Switching to binary was an inspired choice. Now in the infinite hotel, every value between 0 and 1 has meaning when announced by the manager (though that is not true for any of the finite hotels).

But now consider the state of the infinite hotel after either 0.1 or 0.01[...] are announced. If the infinite hotel starts out with every room filled, in both cases, room 1 will be empty, and all the other rooms (all $\infty$ of them) will be occupied. That is to say, announcing 0.1 or announcing 0.01[...] have the same effect on the hotel. They are equivalent as far as the hotel is concerned. (I.e., 0.1 = 0.01[...] in your prposed scheme)

In both cases, the occupant of room 1 must leave room 1. In the case of 0.01[...], they then move into room 2 (of course bumping everybody up), and in the case of 0.1, they just go home. Either way, after the announcement, the hotel now has 1 empty room and an infinity of occupied rooms.

Let me explain in detail why your scheme is flawed. I'll only use a minimum amount of math, promise.

The hotel manager uses the technique of "announcing numbers" for two entirely different reasons. Sometimes, announcing a number removes an occupant from a room and does not specify where they go. Other times, announcing a number shifts occupants to successive rooms. But, imporantly, these algorithms are not the same.

But it's worse than that. Each of these algorithms only works for a tiny set of numbers. Consider a hotel with occupants in just the first two rooms (0.99). Only two rational numbers can be announced which shift occupants to new rooms : 0.891 and 0.081. And only three rational numbers remove occupants without shifting : 0.9, 0.09, and 0.99. But the set of rational numbers is infinite, so this cannot be the operation of subtraction defined for the rationals. For example, announcing 0.783 is meaningless, but subtracting 0.783 is perfectly valid.

You have mistaken the manager's actions for subtraction because your technique of announcing numbers just happens to give the same results as subtraction in these 5 cases. And you still had to give different processes for 0.891 and 0.081 on the one hand and 0.9, 0.09, and 0.99 on the other to force the outcomes to match subtraction.

In order for your technique to correspond with subtraction, you would need to describe a single algorithm that could handle all rational inputs. And then show a contradiction.

Crucially, announcing 0.89[...] and announcing 0.9 in the infinite hotel are NOT identical. One shifts occupants, the other removes the occupant in room 1 without specifying their destination. Interestingly, both leave the hotel in the same state : 0.09[...]

Remember, a broken watch tells the correct time twice a day.

Yes, I also wondered about the reference to Propositional Calculus. Prop Calc would appear to be a purely intellectual pursuit. One might dabble in it entirely sans technology. What better way to while away those boring hours after the fields have all been plowed and the cows milked?

Oh, I see. So you don't believe that an infinite set can be dense. Got it.

We speak different languages.

(And I gotta hit the hay. TIDF, have fun.)

Then this is also not new but the old re-formulation of Xenos Paradox — Deus

Which one? Be specific.

(Hmm, coming up with your own definition of infinity ... is this MU ?)

Yeah, I couldn't figure out how to indicate repeated digits. The underscore also seems to work if we agree on usage.

As with other debates we've had on TPF, we find philosophy majors trying to bring the wishy-washy terminology of philosophy into a math discussion. Infinity is not understood as a math concept, but as a poorly defined common-language concept. :roll:

Let me help : There are an infinite number of rational numbers between 0 and 1. Also between 1 and 2. And so on. An infinity of infinities. And yet Cantor proved that the cardinality of the set of rationals is the same as the cardinality of the integers - they can be placed in a 1-to-1 relationship.

No problem there.

You think the phrase "infinity of infinities" is suspect? How so? Use math please.

And I still don't see how you get from keystone's linear approach (numbers are only added or subtracted) to non-linear systems. Seems like a fair amount of "tweaking".

So where's the wall?

Essentially the use of non-linear equations could be useful. — Deus

Example, please.

I'm using an underscore as the symbol to represent an infinitely repeated digit, as keystone has. Try reading it again.

What "tweaking" of keystone's incorrect approach "solves" the problem?

Also, claiming Hilbert constructed a wall for mathematics is a strange take.

Wow, you're way off base. See my reply to keystone, above.

Again,

0.9 - 0.89 = 0.09 (You don't need 1 on the end)

and

0.9 - 0.9 = 0.09

So, 0.89 = 0.9

You're trying to use "announcing numbers" to stand for two different things : emptying a room into the hallway and shifting occupants to successive rooms. It can't be both.

Under your scheme, announcing 0.9 creates the same problem for a finite hotel as an infinite hotel : any occupant of room 1 is now standing in the hall !

Proof that 0.891 = 0.9 : announcing either 0.891 or 0.9 leaves the infinite hotel in an identical state, namely 0.09

So what's to be done, man?!?

What might be the preferred response to this situation? Does your pessimism allow for action? Or is the disconnect permanent?

Do we return to a pre-industrial, cottage-industry, butcher-your-own-cows existence? Do we strive to put humanity into our tech? Do we look to a "return to nature"? (Most of us would need GPS to even find nature.)

It seems no good if the individual attempts to address the disconnect but society goes on embracing modernity.

I thought the definition of profit was revenue - cost, when revenue > cost. I didn't realize the definition of profit was what you did with it! My bad!

The CEO is unlike other "employees" in that they are over-compensated for their input. They pocket profit created by the labor of others.

Hey, I'm not against capitalism. But it should be tempered, via regulation if necessary.

But I'm not going to get into a debate with a die-hard laissez-faire capitalist. I'm too busy these days - I only look in on TPF sporadically. The floor is yours, sir.

Obviously, profits go into the owners (and shareholders, CEOs) pockets.

In a perfect world (re ), owners are benevolent, and profits are plowed back into the economy and benefit society. And sometimes, they are. Usually however, profits are used to gain more power.

What if we were a species who found working for another individual (or small group) anathema? We just would not do it (I mean right from the get-go, right from the moment our species achieved consciousness). This could either be from the purest form of greed (Perfect Libertarianism) or from the purest form of communalism (Perfect Communism). What kind of society might emerge? Would progress be possible for such a species? Would institutions form?

(Hey, I'm a big lefty. I think the citizens of the US are hopelessly right-wing. US citizens, almost universally, dislike thinking for themselves and are far too trusting of the oligarchs. But, of course, that's a problem for all of humanity, and the US is no better. We have two parties in this country : the far-right R's and the center-right D's. Neither speaks to me. I'd emigrate if it was easy to do - although, I'm not sure where I'd go. If Trump gets back into power, I've taught my children to goose-step and salute and finally turn the old man in as a hopeless lefty to prove their loyalty.)

So there are no truths of math? What, in your wisdom, is math after all?

I assume that - other than simple arithmetic with positive integers (basically, what you can do with your fingers and toes) - you believe mathematics to be made up gobbledy-gook. To live in the modern world and be such a willful know-nothing is breath-taking.

I begin to discern what happened : someone received a low grade in calculus at university and has had it out for those idiot mathematicians ever since.

I didn't say the axioms are wicked, I said they are wlly nilly. — Metaphysician Undercover

Fine, willy nilly then. You miss the point : somehow the truths of math have been revealed to no one else but you.

I ask that you stop being so selfish and share this revealed folk wisdom with the world. Begin here by detailing some misbegotten axioms you have encountered. Then write up your math musings and send them off to prestigious math journals. I am sure they will fight to be first to publish. Your fame and fortune await.

... unlike the undisciplined mathematicians who make willy nilly axioms however they please ... — Metaphysician Undercover

Oh Great Anonymous Forum Poster - who clearly knows better than some of the greatest minds humanity has produced for the past 5000 years - enlighten us : please give an example of these wicked axioms that mislead our youth that we might know them and revile them.

(This should be good.)

Sadly, it seems math discussions on TPF are doomed to descend to the level of farce. Notice that most folks on TPF avoid these topics like the plague. This is most likely because they've seen what happens far too often before.

Consider the latest comments regarding irrationals. Let's draw an analogy. Imagine that every time you saw a topic where consciousness was under discussion, you stepped in and declared that consciousness obviously did not exist because the first three letters of the word spelled "con". How long before you got canned for the low quality of your posts? That's the equivalent of what's happening here.

Some TPF worthies are convinced they know more about math than some of the greatest minds that humanity has produced over the past 5000 years. They know what they know, and there's no arguments that will convince them otherwise. How foolish of us to question the folk wisdom of anonymous forum posters!

Some months ago, on another thread, I implored one of these commenters to write up their math musings and send them off to prestigious math journals. The world languishes without access to these amazing insights. (I don't think he took my advice.)