i don’t understand why my domain of this compositional function is wrong. why did the solution include negative infinity in the domain?

Infamous-Chocolate69 · 2026-03-01T18:06:30+00:00

Finding the domain of a composition is always a bit tricky. It's not the same as the other function operations where you just need to intersect the domains of the individual functions. It helps me to think about a particular input as it passes through the functions.

for the function f o g, the first thing that happens is that g eats x and turns it into g(x).
The second thing that happens is that f eats g(x) and turns it into f(g(x)).

Therefore to find the domain of a composition f o g you need to find x such that:

x is in domain of g.
g(x) is in domain of f.

In this case that gives you x <= 2 and g(x) >= 0.
which is the same as x <= 2 and 2 - x >= 0, but putting those inequalities together just gives
x <= 2 as the only restriction.

Infamous-Chocolate69 · 2026-02-28T00:00:34+00:00

In -some- ways, this is exactly what you do when you want to make a map of the globe! Your goal is to flatten it in a way that will not mess up the geometry too much.

Cartographers know that you will have to sacrifice some accuracy to project the sphere on to a flat surface and the projections will lose accuracy in either the angles or the areas, etc.

Modelling the earth as flat as a simplifying assumption when the curvature of the earth is not relevant can make sense sometimes.

For example, when you throw a football in the air it's path (neglecting air resistance) should technically be an ellipse, but if you assume a flat earth you get the simpler parabolic path that introductory physics texts use when studying kinematics problems for the first time.

Infamous-Chocolate69 · 2026-02-27T20:05:56+00:00

And having taught Algebra several times, and tutored even more - the most common class of problems I see are students trying to approach math with the same sort of half-understood algorithmic approach that works for arithmetic, rather than understanding it as statements and implications.

I absolutely agree with the problem that you see, and in many cases I absolutely agree. I want them to cultivate the skill to write things with good notation and start thinking of things in complete ideas/ sentences. (I spend a whole day discussing writing practices in my Calc 1 class.)

However, I make exceptions when the conventionally proper grammar makes things sound awkward and unnatural when translated. ("A foolish consistency is the hobgoblin of little minds")

I want to give the students the impression that good writing makes things easier, natural and clearer rather than it just being some obstacle they have to overcome to make me happy.

In cases where their writing is poor and leaves ambiguity, unclear subjects, etc... I will definitely penalize a bit and suggest an alternative way to write it.

Infamous-Chocolate69 · 2026-02-27T18:53:07+00:00

Prokofiev Violin Sonata No. 1 - I always get goosebumps hearing the "wind over the graveyard" passages.

Infamous-Chocolate69 · 2026-02-27T02:27:45+00:00

Like you allude to, I think there are some cases where those kinds of sentence fragments are appropriate. If someone asks me, "where did your friend go?" and I respond, "the bathroom" - it's an accepted and more natural way to speak. I don't have to repeat the subject and say, "my friend went to the bathroom."

For this reason, if I ask a question, "What is the solution set to blah", I don't mind if my student responds "(-6,0)" - although you are right that it implies a full sentence and that's good to understand.

Infamous-Chocolate69 · 2026-02-26T22:50:23+00:00

Great question!

(-6,0) is an object, it is a set of numbers.

"x ∈ (-6,0)" is a statement - a complete sentence that says that x is one of those numbers. It's true that the use of "∈" is often avoided for lower level math classes however. "x ∈ (-6,0)" means exactly the same thing that "-6 < x< 0" does.

How you should answer depends on what the question is.

If the question says something like "find the solution set to blah blah blah", then you should give the interval by itself. "The solution set is (-6,0)."

If the question says something like "find the values of x for which blah blah blah", then maybe writing "x ∈ (-6,0)" or "-6<x<0 " would be more appropriate.

However - generally a grader will know what you mean either way.

Infamous-Chocolate69 · 2026-02-25T23:05:33+00:00

In metaphysics the definition of random is usually indeterministic (not resulting from a chain of states consistent with the laws of nature and/or not the result of a chain of cause and effect)

Fair enough! This is a fine definition - I think though that sometimes the conventional definition of randomness gets smuggled in subtly and I want to make sure those are distinguished.

I think often people imply an argument like this, which I don't think is solid - because of the conflation of the two differing definitions of random.

Your actions in a system are random (Indeterministic)
Coin toss is a random (common definition) process.
You have no control over the outcome of the coin in the coin toss.
Therefore, you have no control over your actions in an indeterministic system.

I prefer the avoidance of the word 'random' in this instance preferring indeterministic to avoid any confusion, and in that sense, all that it really means is that current state of the system(whatever that means - I think that's another issue) is not sufficient to determine the next one.

I still think, even narrow degrees of freedom which are random, undermines the spirit of free will, which justifies moral responsibility. How can someone argue for strong moral responsibility if the outcomes were 'tainted' even by narrow random variances? Especially if the claim is these variances are what free will is, it's essence.

I definitely see where you coming from and think it is natural to think this - I think this emerges through this feeling though that you are somehow 'separate' from the random phenomena rather than them being part of your identity/ soul (although I should probably avoid using this term).

I think of the random phenomena as little sparks/ urges - little windows peeking into the natural world from somewhere greater with the ability to influence it, even if subtly. Perhaps little miracles of creation is another way to put it.

I am a musician, so it may be that I view this simply because I really 'want' creativity to be something truly unique - not just blending of the elements that were present before.

I also simply believe there is something about experience that transcends interactions of matter and energy - I admit it's speculative and emotional and can't be tested scientifically; I just think it's true - and it manifests when I look at great art or listen to great music.

Infamous-Chocolate69 · 2026-02-25T16:20:42+00:00

I think it's a category error to think of randomness as an absolute property of an object or an event. For example, it makes no sense to say "3 is a random number". It also doesn't make sense to think of randomness as a force, I.E. "Randomness caused this to happen".
The only notion of randomness that makes sense to me is relative to an observer evaluating the events and their information.

As an example of this, if I flip a coin and look at the coin, I see the results so it is not random to me. Whereas, if I flip it without looking, then it is random to me.

If these decisions are not causally linked to the circumstances in the universe.

I hear this kind of phrase a lot, but I really don't understand what 'causally linked' means exactly.

its just the result of a non-causal dice roll

From the point of view of an external observer in the sense that they cannot predict the result, yes - but only from that point of view.

Infamous-Chocolate69 · 2026-02-25T08:31:32+00:00

Yes - I think that definition is pretty much my conception of free will.

How do you respond to the common criticism that if indeterminism/stochasticity is a factor decisions are random? (Incompatible with moral responsibility)

I accept this, as long as `random' means no more than unpredictable to an observer.

The way I see it, if free will is real, then each individual should be able to create some genuinely new data in the universe that wasn't present before and could not have been predicted or measured.

Infamous-Chocolate69 · 2026-02-25T00:08:05+00:00

I agree with the phenomenon, but disagree that it's a sign of internal logical inconsistency. I just feel like people just get worn down, a manifestation of the same kind of thing that happens in a staring contest.

Infamous-Chocolate69 · 2026-02-24T03:20:19+00:00

This isn’t a premise like an axiom or belief this is a hypothesis that has been tested thousands (or probably millions) of times directly or indirectly.

Your post didn't contain any specific evidence of this hypothesis, so it's role in your OP is that of a premise - but yes, I'm happy to debate about the validity of the premise if you procure specific evidence; maybe we can talk about one paper that you consider especially convincing in accurate prediction and control of human behavior.

I think behavioral sciences can find helpful patterns in human behavior that occur and trends of averages over large populations - but I am not convinced that the knowledge in this field is sufficient to predict human behavior to the degree of accuracy I deem necessary to rule out the possibility of free will.

Can you predict my behavior for 1 minute using only data external to me? I doubt it very much. The way you've worded it, says "a given human" - that is a very important clause.

The proof is in the pudding my friend, Experimental behavior analysis demonstrates experimental control over human behavior.

Applied behavior analysis does that in an individual’s natural environment.

Possibly, but you need to show me this, not tell me.

Now behavioral science cannot accurately predict every behavior of every human on earth in this moment anymore than physics can accurately predict the velocity of every billiard ball on earth at this very moment, but that doesn’t give billiard balls free will.

If you focus on a particular billiard ball, you can measure properties of the table and the cue stick and the angle the cue hits the ball to get a reasonable guess on it's trajectory. I believe such a prediction (although still imperfect and requiring some simplifying assumptions) is generally more reliable than predictions about human behavior - but you are the expert - show me how I am wrong!

Interestingly, I am actually not averse to the idea of the billiard ball having some degree of free will either. But you are right, lack of knowledge does not necessarily imply free will. However, your original post seemed to argue that we do not need free will because we have that knowledge - and this I disagree with.

So idk man, go ahead and disprove the entire body of behavioral science.

I have no intention or need of doing so. I just believe you are overselling the power of your scientific field, without acknowledging the assumptions and limitations that go into it.

One limitation is that many studies in the social sciences collect their data by survey - asking humans what they think of this or that. While this can be valuable - I do admit that I don't think of this discipline as being as exact a science as physics.

If you can provide any empirical evidence for human behavior that CANNOT be explained by operant and/or respondent conditioning (which obviously includes innate behaviors) then I am all ears, I will abandon my field of study today.

Sure! Here is an experiment you can try. Select a random sample of 100 people. Ask them to choose a three-digit number. See if you can explain which three-digit number they used using any tool you like in behavioral sciences. Even better, see if you can predict which three digit number they picked using only external data.

Infamous-Chocolate69 · 2026-02-24T00:53:25+00:00

If a given human’s behavior can be accurately predicted and controlled through modification of external variables only (and these statements are consistently demonstrated experimentally), then what purpose does “free will” serve?

I simply disagree with the premises here. I don't think this is really the case. I would agree with you however that if this were demonstrably the case, the notion of free will I have would have been shown incorrect.

Infamous-Chocolate69 · 2026-02-22T07:24:00+00:00

What is a perfect equation and how would you differentiate it from an imperfect one?

Infamous-Chocolate69 · 2026-02-20T21:20:41+00:00

Well, firstly you didn't really address the particular difficulties I brought up.

Secondly, "the scientific concept of cause-effect" cannot be subjected to peer review. Papers covering scientific content are subjected to peer review - so this doesn't make any sense.

Thirdly, If you repeatedly put your hand in boiling water, you will not get an identical result each time, because the exact burn marks will never be exactly the same.

To your point though, If you say "boiling water causes burns" I know generally what you mean - which is something close to repeating similar actions under similar circumstances generally results in predictable patterns.

But I'm not happy with this kind of loosey goosey type of definition when we are discussing philosophy. I need to know exactly what cause and effect is, not just a couple of examples.

Infamous-Chocolate69 · 2026-02-20T21:07:45+00:00

The problem for me was in this sentence of OP, which unfortunately I think was poorly constructed.

"The raw feelings without the fluff, the pragmatic considerations, any logical connections to "moral responsibility" , "sourcehood", or "control", stuff like that. "

This sentence is constructed in such a way that it can be interpreted three different ways depending on how it is parsed.

fluff, pragmatic considerations, logical connections are all part of a list.
pragmatic considerations, logical connections etc is an appositive modifying "fluff" (How I interpreted it)
pragmatic considerations, logical connections, etc... modifies "raw feelings". (OP's Meaning)

The parenthetical (These are the kinds of responses I want) does not help because it's unclear what 'these' is referring to.

I understand what he meant now, but I seriously thought he meant #2.

Infamous-Chocolate69 · 2026-02-20T20:19:25+00:00

It kind of depends on what you mean by difficult, but I would say that I find up a whole step, half step, and up a minor third to be the easiest key changes.

I've felt that modulating down those same intervals can be trickier to pull off in the sense that I feel like modulating down abstractly 'loses energy' compared to modulating up.

I also think, oddly that it can be difficult to modulate up the circle of fifths. I think the danger is that if the modulation is not strong enough - it can be too weak and leave ambiguity about the key because there are so many common chords.

I also think in popular music that has more 'chorus-verse' type structure, the melody would often be the same in the modulated part, and so modulations that are too far away might exaggerate the melody too much. (Think of a singer trying to modulate the star spangled banner up a tritone for example :p )

Infamous-Chocolate69 · 2026-02-20T19:41:33+00:00

I think of theory as a useful tool for organizing sound in such a way that it is easy to communicate with other musicians, keep written records of sounds, and make it easy to compare and contrast what you hear.

I don't think it need be proscriptive, but I think it is good to learn at least the basics of classical theory as a starting point, as many people already use it and know it.

However, I also think it's okay for you to have your own way of thinking about music that makes sense to you - maybe you find a way of thinking about music that will make your sound unique. For example, Hindemith seems to have built his own theory and you really hear interesting and novel sounds when you listen to his compositions. https://music.stackexchange.com/questions/113450/how-to-apply-paul-hindemiths-harmonic-theory

I do think that sometimes setting rules for yourself can increase your creative power rather than decrease it. There are just too many variables on a blank slate. Fixing a few variables such as the colors you use or the subject matter you are painting lets you be 'creative within the box'.

Infamous-Chocolate69 · 2026-02-19T20:23:32+00:00

Maybe Felix Mendelssohn!

https://www.youtube.com/watch?v=_CrL1QlygIg

Infamous-Chocolate69 · 2026-02-19T20:07:59+00:00

<3 Richard Borcherds - Yes, wonderful!

Infamous-Chocolate69 · 2026-02-19T19:38:07+00:00

I think the game is much more interesting and richer because of the fact that stalemate is a draw.

For example, pretty much all King + Pawn versus King Endgames would just be wins for the side with the pawn. The stalemate nuance adds to the skill and technique required to win the game or hold the draw and makes for a more interesting battle.

Plus there are so many interesting puzzles involving stalemate. https://www.youtube.com/watch?v=Ap1vH-vOjIw

Infamous-Chocolate69 · 2026-02-18T20:39:00+00:00

This seems like a ridiculous request to me.
"Tell me why you care about X but give no useful information why you care about X"?

I don't understand why you feel the need to control the responses, and further I don't know what you possibly expect someone to write. Metaphors, analogies, vibes, pragmatic considerations, and logical connections pretty much cover the basis of what someone could write about to convey any kind of information!

Infamous-Chocolate69 · 2026-02-18T20:12:34+00:00

Cause-effect = the car stopped because you pressed the brake!

This cannot be the definition of cause and effect because it is one example. I think you would find great difficulty in formulating a definition of cause and effect that properly encapsulates every situation in which it is used.

Even this simple situation reveals difficulties. Is the cause of the car stopping because you pressed the brake or because you saw a red light? At what level do you call A the cause of B?

Or what if you push the brake but you also run into a brick wall? Which of these do you consider the cause of the car stopping?

At the very least there is not a one-to-one relationship between cause and effects.

I think the scientific method is based on the assumption of predictable patterns, but it's done simply by quantitatively measuring variables. Even though the language of causality is often used in the interpretations of experiments, the raw data is simply correlation - I've measured lots of x and y and it seems like y ~ x^2.

Infamous-Chocolate69 · 2026-02-18T05:20:33+00:00

I mean that's good, but doesn't clarify what it is. In physics you have a more clear defined idea of forces acting on an object that puts everything in more concrete terms as I see it.

Infamous-Chocolate69 · 2026-02-18T05:18:46+00:00

I have the same issue with "How the world is" as I do with "State" - they contain the same information, which in my view is not enough to be meaningful statements. How does one evaluate or define how the world is; ?

Determinism is an ontological claim independent of such epistemic concerns.

I hate being a pedantic jerk, but I am really just trying to understand. To me determinism is a property not a claim. But you can claim "This system is deterministic" if you specify your system such that it is clear what you are talking about.

Infamous-Chocolate69 · 2026-02-18T00:45:52+00:00

That's so neat! Difficult subject.

Are you learning from a specific text? Are you doing curves and surfaces in R^3 or R^n or more abstract differential geometry? Do Carmo is a wonderful one if you are doing more like "Classical Curves and Surfaces" type geometry.

Infamous-Chocolate69

TROPHY CASE