Beat Unknown Soldier at his own game of math.

[Juvenal]

Lowest possible value (x - 1)^2 can have is zero, if x = a/b or b/a, regardless the values of a and b, as long as they are real numbers.

So the solution is \(x=1\). Now why is that a problem?

 

[Unknown Soldier]

You jumped into a conversation tat goes back to sildier’s thread ‘Why mathematics is neither absolutely nor objectively "right."’

He clams that 2 + 2 does not alwys equal 4 in math, therefore math us not absulute and is open to subjunctive interpretation.

That is correct. I proved my case in the OP of that thread.

As a math prof you may want to read through that thread.

That's a very good idea. Anybody interested in the truth value of math should check out that thread.

Soldier says he read books on math and that qialifies him as a mathematician.

Yes I study math books--a LOT of math books. I recommend you study a math book some time.

We reacted to his bogus logic as he tries to come up wth clever problems to amaze us, all the while not understanding fundamentals.

You didn't understand my logic. You're inability to grasp the subject matter doesn't make my proof wrong.

As to busting the professor, why would I want to do that? My skill was synthesizing systems, not math trivia. I was never bored trhroughout my adult life.

There's nothing trivial about the math I've presented, but I'm not surprised that you might think it's trivial.

 

[steve_bank]

Still looking for attention Soldier?

When I was working once a year I'd go through my math and other texts working problems as a review. Havenn't opened a bok in going on 7 years now, so I am a bit rusty.

Soldier, what you are posting I take as pretty ordinary stuff, what I call plug and chug. If you know the theory the answer is mechanistic. I could throw out a number of maths you are not likely to comprehend. Of importance to EEs, in electric circuits and electromagnets what is the meaning of zero, positive, and negative divergence? What is the significance of the Divergence Theorem? Convolution? Applying math requires mire the wrote skill.

A common thread across disciplines is curl and divergence. From a math perspective there is no difference between fields in electromagnetics and fluid mechanics. The stiff you post would have been utterly useless to me.

Here is as simple a kind of engineering math problem there is.

t = time
v = 12
r = 1000
vr(t) = V - v(t)
v(t) = V - vr(t)
c = 1e-6
i = c *dv/dt
I'll give you i(t) =( V - v(t))/r
V - vr(t) - v(t) = 0
Solve for v(t).

A simple mechanics problem.
a = acceleration
s = distance
t = time
a(t) = (4 + t)^2
Initial conditions s = 0 t = 0
What is the distance traveled between t = .2 and t = .4?

A problem I remember from my Multivariable Calculus text.

You have a pyramid. What is the equation of a line from the center of the base to the peak of the pyramid?

Juvenal aooears to be who he says he is, no doubt he can solve it.

 

[Juvenal]

\( x+1/x \ge 2 \to \\ (x-1)^2 \le 0\) because \(x<0\)

This has the unique solution \(x=1\), so the corollary is true on this restricted domain. Now here's the kicker. This also includes an error.

Find it.

From the above, we have the contradiction \(x=1\) because \(x<0\).

 

[Juvenal]

Still looking for attention Soldier?

When I was working once a year I'd go through my math and other texts working problems as a review. Havenn't opened a bok in going on 7 years now, so I am a bit rusty.

Soldier, what you are posting I take as pretty ordinary stuff, what I call plug and chug. If you know the theory the answer is mechanistic. I could throw out a number of maths you are not likely to comprehend. Of importance to EEs, in electric circuits and electromagnets what is the meaning of zero, positive, and negative divergence? What is the significance of the Divergence Theorem? Convolution? Applying math requires mire the wrote skill.

Curl and divergence show up in Maxwell's equations. Convolution shows up in Laplace and Fourier transforms, and more conventionally in computing the products of power series.

A common thread across disciplines is curl and divergence. From a math perspective there is no difference between fields in electromagnetics and fluid mechanics. The stiff you post would have been utterly useless to me.

Here is as simple a kind of engineering math problem there is.

t = time
v = 12

I assume you meant \(V = 12\)

r = 1000
vr(t) = V - v(t)
v(t) = V - vr(t)

It looks like you're trying to describe a simple series RC circuit powered by a 12 volt battery, with the voltage broken into components across the resistance load \(v_r(t)\) and a capacitor \(v_c(t)\).

c = 1e-6
i = c *dv/dt
I'll give you i(t) =( V - v(t))/r
V - vr(t) - v(t) = 0
Solve for v(t).

Which would mean you're asking for the voltage across the capacitor \(v_c(t)\).

A simple mechanics problem.
a = acceleration
s = distance
t = time
a(t) = (4 + t)^2
Initial conditions s = 0 t = 0
What is the distance traveled between t = .2 and t = .4?

This one's just weird.

Generally, you have constant acceleration from a constant force, but here the acceleration, hence the force, is increasing as the square of the elapsed time.

Now the position function is going to require two initial conditions and the speed function will require just one.

\(\displaystyle a(t)=(4+t)^2, v(t) = \frac{(4+t)^3}{3} +v_0, s(t)=\frac{(4+t)^4}{12} +v_0 t + s_0\)

But because you're looking for \(\Delta s\), the one initial condition given, \(s_0\), is irrelevant, and the one needed, \(v_0\) is missing.

A problem I remember from my Multivariable Calculus text.

You have a pyramid. What is the equation of a line from the center of the base to the peak of the pyramid?

Juvenal aooears to be who he says he is, no doubt he can solve it.

Looks like you're probably assuming a right square pyramid, so the line would be given parametrically by the center point of the base and a direction vector.

\( \bar{r}(t) = C +t\bar{v}\)

If you know the four corners of the base, the center is simply their average

\(\displaystyle C = \sum_{i=1}^4 B_i/4\)

and a direction vector can be computed from vectors linking any one corner to two others

\(\bar{v} = \overline{B_1 B_2} \times \overline{B_1 B_3}\)

 

[steve_bank]

On the first problem by imnsection it is a 1st order ordeinaty differntial equation. Seeing that the transient response to a steady state input , unit step, is exonential. It is a matter of finding the constants.

Plenty of RC ciruit examples on the net.

This one is trviall but on a more complex circuit Idd solve it numerically. As an example imleted t as a macro.

An example of time domain convolution. In te copex frequncy domin it would be (1/S)(1/(SRC+1), Convolving the unit step with a trader function. Take the inverse LaPlace Transform to get the step response.

'Convolution in the time domain is multiplication in the S domain'.

Sub Main

dim N,j as integer
dim dt,_time,vc,R,C as double

N = 800
dt = 1e-4
_time = 0.
Vc = 0.
R = 1000
C = 10e-6

dim t(N),y(N),i(N),Vout(N) as double

' time base
for j = 1 to N
t(j) = _time
_time = _time + dt
next

' unit step input signal
for j = 1 to N
y(j) = 1
next j
for j = 1 to 200
y(j) = 0
next j

for j = 1 to N
i(j) = (y(j) - Vc)/R
dv = (dt*i(j))/C
Vc = Vc + dv
Vout(j) = Vc
next j

Dim Doc As Object
Dim Sheet As Object
Doc = ThisComponent
Sheet = Doc.Sheets (0)

' zero cells
for j = 1 to N
Cell = Sheet.getCellByPosition(0,j)
Cell.Value = 0
Cell = Sheet.getCellByPosition(1,j)
Cell.Value = 0
Cell = Sheet.getCellByPosition(2,j)
Cell.Value = 0
next j

Cell = Sheet.getCellByPosition(0,0)
cell.string = "TIME"
Cell = Sheet.getCellByPosition(1,0)
cell.string = "V OUT"
Cell = Sheet.getCellByPosition(2,0)
cell.string =" CURRENT"

for j = 1 to N
Cell = Sheet.getCellByPosition(0,j)
Cell.Value = t(j)
Cell = Sheet.getCellByPosition(1,j)
Cell.Value = Vout(j)
Cell = Sheet.getCellByPosition(2,j)
Cell.Value = i(j)
next j

End Sub


For the second problem

s = distance mters
t time seconds
v velocity ds/dt
a acceleration dv/t

Acceleration us the 2nd derivative. To get velocity integrate acceleration and so on.

a(t) = (7 + t)^2
= 49 +14t + t^2
integrating ... 49t + 7t^2 +t^3/3
I am out of practice but I thing that is it. Integrate once more time to find distance traveled.

I was poking at Soldier.

It took a few years on the jobfor it to fully sink in. The math is the same across diclines wit different tags and jargon. Aesonat elecrical LC circuit and a shock absorber-spring suspension on a are both 2nd order equations.

Divergence of a volume tells whether the volume is a source, sink, or passive. Divergence is an expression of conservation.

The divergence of the volume of a battery and a water pump will indicate a source.

For an incomprehensible fluid in a pipe with trbukent fkow the dibegence of any volume in the flow should be zero indicating input to the volume equals output.

The importance of the Divergence Theorem is converting a surface integral into the suumation of small volumes withing the surface as dydxdz goes to zero if I remember right. As an example in electrostatics charge enclosed in a box has problems with integrating the surface at the edges and corners, discontinuities. Divergence Theorem resolves the computational problem.

More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the "flux" through the surface, is equal to the volume integral of the divergence over the region inside the surface.

As I said I was not a mathematician, but I had to have some depth and understanding to be able to apply it and to be able to read books.

Soldier, as you are a math book reader a recommendation. There should be cheap old editions, it goes back quite a ways. I kept it at work as a reference.

Basic Technical Mathematics with... book by Allyn J. Washington

 

[Juvenal]

a(t) = (7 + t)^2
= 49 +14t + t^2
integrating ... 49t + 7t^2 +t^3/3
I am out of practice but I thing that is it. Integrate once more time to find distance traveled.

You've omitted the constant of integration, \(v_0\), which is needed to calculate \(\Delta s\).

And while it's not an actual error, there's no point multiplying out \((7+t)^2\) or \((4+t)^2\) as originally stated.

Let \( u=7+t\)

Then \(\displaystyle \frac{du}{dt}=1, du=dt\)

and \(\displaystyle \int (7+t)^2 dt = \int u^2 du\)

 

[Jimmy Higgins]

\( x+1/x \ge 2 \to \\ (x-1)^2 \le 0\) because \(x<0\)

This has the unique solution \(x=1\), so the corollary is true on this restricted domain. Now here's the kicker. This also includes an error.

Find it.

From the above, we have the contradiction \(x=1\) because \(x<0\).

Your problem is that it is never less than zero at all. Regardless the value of x. The equation is wrong.

 

[Swammerdami]

You've omitted the constant of integration,

Two mathematicians were out drinking one night, lamenting the state of Americans' math education. Chuck went to the bathroom. Dan hastily wrote "one-third x cubed" on a napkin and motioned to the cocktail waitress. "Memorize this and say it when I ask you a question" he said while handing her a banknote.

Chuck returned and signaled for another round of Jack Daniels. Dan said "I think you're underestimating us Americans. I'll bet that buxom blonde cocktail waitress even knows some math." When she arrived he asked her "What's the integral of x squared?" She recited "one-third x cubed" and Chuck was suitably impressed.

But the waitress glared at Dan and said condescendingly "Plus a constant!"

 

[Juvenal]

You've omitted the constant of integration,

Two mathematicians were out drinking one night, lamenting the state of Americans' math education. Chuck went to the bathroom. Dan hastily wrote "one-third x cubed" on a napkin and motioned to the cocktail waitress. "Memorize this and say it when I ask you a question" he said while handing her a banknote.

Chuck returned and signaled for another round of Jack Daniels. Dan said "I think you're underestimating us Americans. I'll bet that buxom blonde cocktail waitress even knows some math." When she arrived he asked her "What's the integral of x squared?" She recited "one-third x cubed" and Chuck was suitably impressed.

But the waitress glared at Dan and said condescendingly "Plus a constant!"

Yes, it's a well-worn joke, but it actually happened one night in Chicago at Giordano's on Rush where I was working as upstairs host.

In the true life version, it was a waitress trying to get a beat down on another waitress by using a table of customers in a booth, the integrand was just x, the answer given to the table was x^2/2 and the, gal at the table answered completely, "x^2 / 2 plus a constant of integration."

I was 10 feet away from the booth and heard it all, and damn near pissed my pants laughing when the gal in the booth piped up with that answer getting full reactions from BOTH waitresses!

 

[Juvenal]

\( x+1/x \ge 2 \to \\ (x-1)^2 \le 0\) because \(x<0\)

This has the unique solution \(x=1\), so the corollary is true on this restricted domain. Now here's the kicker. This also includes an error.

Find it.

From the above, we have the contradiction \(x=1\) because \(x<0\).

Your problem is that it is never less than zero at all. Regardless the value of x. The equation is wrong.

Some day somebody is going to convince you that nouns and descriptive adjectives are your friends. What is this "it" that's never less than zero? And no, there's nothing wrong with an equation that includes a contradiction. That's how indirect proofs are done, in this case, the indirect proof that there's no solution for the given inequality if a and b don't share the same sign.

 

[Juvenal]

You jumped into a conversation tat goes back to sildier’s thread ‘Why mathematics is neither absolutely nor objectively "right."’

I finally got a chance to finish reading that thread, and it's a full-on, OCD-triggering horror show for any pure mathematician.

Every single post purporting to demonstrate the objective truths of mathematics was wrong. And every post purporting to demonstrate that mathematical statements are not objectively true was also wrong.

The only relaxing moment was a quote brought in by an avowed non-mathematical poster from a theoretical physicist explaining why pure mathematicians find applications of the mathematics we discover generally irrelevant.

As a non-mathematician I hesitate to post in this thread, but I came across the below quote this morning and I thought it might be of interest. I assume most posters in this thread would consider the observation to be trivially true.

The quote is from The Great Paradox of Science by theoretical physicist Mano Singham, and it is part of a discussion about the difference between pure mathematics and mathematics as used by scientists. Emphasis is in the original.

Even a statement such as “1+1=2,” which most people might regard as a universal truth that cannot be denied, is seen by them [pure mathematicians] as merely the consequence of certain starting assumptions, and one cannot assign any absolute truth value to it. So pure mathematicians concern themselves more with the rigor of proof, and less with whether the theorems resulting from them have any meaning that could be related to truth in the empirical world. What is important is that the axioms be consistent, or at least appear to be so since we can never prove them to be so. Whether they say anything about the physical world that can be described as true has ceased to be determinative.

… So while in mathematics the statement “1+1-2” is simply a string of symbols representing a theorem based on a particular set of axioms and rules of logic, in science, its empirical truth or falsity is extremely important and is judged by how well real objects (apples, chairs, etc.) conform to it.

Rather than “right” Singham uses the term “true,” which I prefer, but perhaps there is a subtle difference that I’m not catching.

There's a quote regularly trotted out by the evangelical Christian apologist William Lane Craig in his regrettable attempts during public debates to resurrect the Kalam Cosmological Argument, citing a famous mathematician whose name I can't recall at the moment, "Infinity doesn't exist in the real world!"

With feeling.

Or something similar. Found a video presentation just now, but it's from a different iteration, and excludes the wanted cite.

The first time I heard the argument from Craig, I concluded that either he doesn't believe in an infinite God or that his God doesn't exist in the real world. Evangelical apologists can have perfect teeth, apparently, but cogent arguments escape them.

What Craig missed was that for an actual mathematician — with apologies to my applied colleagues — while it's certain that infinity doesn't exist in the real world, we could just as certainly say the same thing about the number "2."

There is no instance of this number anywhere, and never will be outside the abstract realm, because it requires identical objects to exist, and they don't. Even the most similar objects that could be brought in as an examples will differ in their location in space and time, at the very least.

The number 2 "exists" only in the sense that thoughts exist.

That doesn't mean that the number 2 doesn't have properties that can be described consistently.

It's not true that 2 + 2 is absolutely 4. Depending on the rules mathematicians are using, 2 + 2 = 0 might be the case. In modular arithmetic, 2 + 2 might not even be defined at all much less true. For example, in binary there is no 2.

Or inconsistently, as the above example shows.

Changing the representation of a number doesn't change the number. Just because a digit doesn't exist in a chosen base doesn't mean that the number represented by that digit in another base stops existing. Neither is the "example" of changing the base an example of modular arithmetic.

In modular arithmetic, the residues are infinite equivalence classes. Had US bothered to ask his online calculator to evaluate 4 modulo 2, he'd have discovered that it is also equal to 0.

Indeed, modulo 2, they are both members of the residue class \(\{0, \pm 2, \pm 4, \ldots\}\).

The power of modular arithmetic is that the results are consistent independent of which element of the residue class is chosen. For example:

Because \(-3\cdot 2 \equiv -6 \equiv 1 \mod 7\), we have that \(\displaystyle \frac{1}{2} \equiv -3 \equiv 4 \mod 7\) so \(\displaystyle 4x \equiv -3x \equiv \frac{x}{2} \mod 7\)

So modulo 7, to cut something in half, one can simply multiply it by either 4 or -3, a result that's not possible if we remove the interchangeability of residues in the same equivalence class.

All of the above could be rewritten in binary without changing the result.

And history suggests US would still manage to muff the explanation.

 

[Unknown Soldier]

It's not true that 2 + 2 is absolutely 4. Depending on the rules mathematicians are using, 2 + 2 = 0 might be the case. In modular arithmetic, 2 + 2 might not even be defined at all much less true. For example, in binary there is no 2.

Changing the representation of a number doesn't change the number. Just because a digit doesn't exist in a chosen base doesn't mean that the number represented by that digit in another base stops existing. Neither is the "example" of changing the base an example of modular arithmetic.

Why, you can make "2" anything you want it to be! The sky won't fall if you do. Likewise, you can have 2 + 2 = 4. or 2 + 2 = 0, or 2 + 2 = The Cat's Meow. There's no law of nature saying you can't. People make all this stuff up. Sure, what math people make up may prove useful, or fashionable, or impressive. And for that matter much of what math people invent can cause great consternation for some people on internet forums when some guy comes along and tries to tell them it's invented. But no matter how much they protest calling that revelation things like "a full-on, OCD-triggering horror show for any pure mathematician," it doesn't change the fact that math is neither absolute nor is it objective.

That's a fact.

As for me, I can live with invented, subjective, and relative math. It's still elegant, fun, and very challenging. Heck, I study it hours a day--every day. And I've been learning it in school and through self-study for decades. After all that hard work, I can go online to see what others think of it. And no surprise--I see they've made it into an idol.

So some people just can't live without a God. If they can't swallow the anthropomorphic God of conventional religion, then they may turn to mathematics as they seek absolute, objective truth.

But there they find unbelievers too!

 

[steve_bank]

a(t) = (7 + t)^2
= 49 +14t + t^2
integrating ... 49t + 7t^2 +t^3/3
I am out of practice but I thing that is it. Integrate once more time to find distance traveled.

You've omitted the constant of integration, \(v_0\), which is needed to calculate \(\Delta s\).

And while it's not an actual error, there's no point multiplying out \((7+t)^2\) or \((4+t)^2\) as originally stated.

Let \( u=7+t\)

Then \(\displaystyle \frac{du}{dt}=1, du=dt\)

and \(\displaystyle \int (7+t)^2 dt = \int u^2 du\)

I am posting informally on an informal forum. If you want to be pedantic knock yourself out. Neither here not there to me.

These are after all simple problems and I am not delivering a lecture. And I am not interested in learninh Latex.

 

[steve_bank]

There was the mathematician who after reading the low probability of a plane crashing due to a bomb on board started carrying a bomb when he traveled on an airplane.

 

[Juvenal]

Why, you can make "2" anything you want it to be! The sky won't fall if you do. Likewise, you can have 2 + 2 = 4. or 2 + 2 = 0, or 2 + 2 = The Cat's Meow. There's no law of nature saying you can't. People make all this stuff up. Sure, what math people make up may prove useful, or fashionable, or impressive. And for that matter much of what math people invent can cause great consternation for some people on internet forums when some guy comes along and tries to tell them it's invented. But no matter how much they protest calling that revelation things like "a full-on, OCD-triggering horror show for any pure mathematician," it doesn't change the fact that math is neither absolute nor is it objective.

Discovered, not invented, or made up.

And again, mathematicians don't care if the mathematics they discover is useful. As it turns out, surprisingly enough, it is, but that's irrelevant to the underlying abstractions that we work with.

That's a fact.

Certainly it's a fact that math isn't objective and is no more absolute than the arbitrarily accepted axioms upon which theory is based.

But it's not because two can't be represented in binary or because the residues in modular arithmetic aren't equivalence classes or because of any other contrafactual you've somehow absorbed and can't seem to keep yourself from sharing.

It's one thing to claim something is true. It's another thing to support that claim. Congratulations on the former. If only you'd stopped when you were ahead.

As for me, I can live with invented, subjective, and relative math. It's still elegant, fun, and very challenging. Heck, I study it hours a day--every day. And I've been learning it in school and through self-study for decades. After all that hard work, I can go online to see what others think of it. And no surprise--I see they've made it into an idol.

Read all the books you like. I commend autodidactism.

But no, that's not you.

There's no benefit in reading a book you can't understand, and less than no benefit when your lack of comprehension causes you to learn things that are not true. That's the time to seek help from reliable sources who can steer you away from some of the bizarre claims you've been making.

That's not personal. It's something that happens to all of us. I had a whole chapter of my dissertation wiped because one member of my committee noticed I'd headed off on a bridge to nowhere. The point being that without independent critical examination, there are no guard rails, and without guard rails, running off the cliff is a question of when, not if.

So some people just can't live without a God. If they can't swallow the anthropomorphic God of conventional religion, then they may turn to mathematics as they seek absolute, objective truth.

But there they find unbelievers too!

There are sensible alternatives to creator-based religions.

Buddhism requires no gods at all. Myriad Hindu sects give no more than a nod to any creator deity. On a hill overlooking Athens are the remains of the Parthenon, a temple devoted to the worship of the goddess of wisdom itself. Within the most prestigious academic degree is a theophoric reference to the goddess of skill and cleverness.

But there's no sensible alternative to talking sense.

If you're going to speak of modular arithmetic, you must at the very least note that you can't cram an infinite number of integers into a finite set of residue classes without making those residue classes infinite as well.

You cant force a number that is abstractly defined by its 1-1 correspondences into the straight jacket of its binary or decimal representation.

And you can't simply bull through explanations of why these things don't make sense without interacting with the criticisms and expect your conduct will not result in the obvious consequences.

Make an attempt to make the time I've spent explaining this to you worthwhile, or lose the benefit of my help.

No skin off me. I've got plenty of other students to keep me busy.

 

[Juvenal]

I am posting informally on an informal forum. If you want to be pedantic knock yourself out. Neither here not there to me.

These are after all simple problems and I am not delivering a lecture. And I am not interested in learninh Latex.

Umm, wow. Dude, you cut off the part that you need to get the solution to the problem that you posed.

Good luck with that. And have a nice day.

 

[Unknown Soldier]

Why, you can make "2" anything you want it to be! The sky won't fall if you do. Likewise, you can have 2 + 2 = 4. or 2 + 2 = 0, or 2 + 2 = The Cat's Meow. There's no law of nature saying you can't. People make all this stuff up. Sure, what math people make up may prove useful, or fashionable, or impressive. And for that matter much of what math people invent can cause great consternation for some people on internet forums when some guy comes along and tries to tell them it's invented. But no matter how much they protest calling that revelation things like "a full-on, OCD-triggering horror show for any pure mathematician," it doesn't change the fact that math is neither absolute nor is it objective.

Discovered, not invented, or made up.

You asserted it. Now prove that assertion. You'll need to demonstrate that math exists prior to its discovery.

I'd say that math is essentially invented. Laplace Transforms were invented by Laplace, and the Cartesian Coordinate System was invented by Descartes. We can see math being invented throughout the history of mathematics by many different people. All that math wasn't just lying around waiting to be dug up. So the knowledge that math is invented is based in the practice of basing mathematics in axioms. Axioms are arbitrary rules that people make up. As such, they are the product of human ingenuity and creativity (i.e. inventions). I've proved the role of arbitrary axioms in mathematics in my What proof is there that 2 + 2 = 4? thread.

That said, there is a discovery of sorts in math in which once some idea is invented, later on that idea leads to unforeseen conclusions. Circles, for example, were invented but later on the number π was found to be the ratio of the circumference of any circle to the measure of its diameter. Nobody including the inventors of circles expected π.

And again, mathematicians don't care if the mathematics they discover is useful.

That's flat-out false. Newton, for example, did some major work in developing calculus because he needed it to do his work in physics.

As it turns out, surprisingly enough, it is, but that's irrelevant to the underlying abstractions that we work with.

That's a fact.

Certainly it's a fact that math isn't objective and is no more absolute than the arbitrarily accepted axioms upon which theory is based.

Right! So you get it. Why are you arguing with me?

But it's not because two can't be represented in binary...

The conventional representation of 2 in binary is 10. Somebody cooked that up, of course.

...or because the residues in modular arithmetic aren't equivalence classes or because of any other contrafactual you've somehow absorbed and can't seem to keep yourself from sharing.

I do like to share my knowledge, that's true.

It's one thing to claim something is true. It's another thing to support that claim. Congratulations on the former.

But I do support my claims. By contrast, your earlier claim that math is discovered is completely unsupported. The kettle calls the pot black!

If only you'd stopped when you were ahead.

I can understand why you want me to stop.

As for me, I can live with invented, subjective, and relative math. It's still elegant, fun, and very challenging. Heck, I study it hours a day--every day. And I've been learning it in school and through self-study for decades. After all that hard work, I can go online to see what others think of it. And no surprise--I see they've made it into an idol.

Read all the books you like. I commend autodidactism.

I'll keep reading books. I recommend you read some books too.

But no, that's not you.

What's not I?

There's no benefit in reading a book you can't understand, and less than no benefit when your lack of comprehension causes you to learn things that are not true.

I'll let you know if that ever happens!

But honestly, many of the books I study can be hard for me to understand. But I would be an idiot to take your advice and stop studying them for that reason! As I see it, if I study ten new concepts, and I only understand one of those concepts, then I've learned one concept.

In any case, self-study has been very beneficial for me. When I was in college I prepared for many of my courses by studying beforehand. I ended up with a four-year degree and a 4.0 GPA.

That's the time to seek help from reliable sources who can steer you away from some of the bizarre claims you've been making.

You're making one of the biggest goofs in mathematics here: You are relying on intuition and rejecting whatever seems strange to you. Many truths in mathematics as well as science are often counterintuitive. Truth doesn't care if it makes sense to us.

And those "bizarre claims" you mention are all based in conventional mathematics and logic. I'm not making up anything.

That's not personal. It's something that happens to all of us. I had a whole chapter of my dissertation wiped because one member of my committee noticed I'd headed off on a bridge to nowhere. The point being that without independent critical examination, there are no guard rails, and without guard rails, running off the cliff is a question of when, not if.

I can only wonder what they would think of your latest post.

I've got plenty of other students to keep me busy.

Send them my way. I've worked as a math tutor. They may well need one.

A great video to watch that explains my position on mathematics is Philosophical Failures of Christian Apologetics, Part 4: Word Games. Note that Christian apologists see mathematics in a way that is similar to the way you see it.

 

[steve_bank]

Juvenal,

Periodically people appear on the forum claiming math and science have it all wrong and try to prove it. Usually Christians defending creationism but not always.

Quantum mechanics proves there is life after deatg.
A theoretical mathematical infinite decimal number line proves the universe is infinite.

Trying to make a rational logical argument never works.

Those who think logically, mathematical nd scientifically are a small minority.

I spent most of my as adult life around people who spoke science and math. Way back when I joined the forum it was a bit of a wake up call.

Keep the faith professor.

 

[Juvenal]

Why, you can make "2" anything you want it to be! The sky won't fall if you do. Likewise, you can have 2 + 2 = 4. or 2 + 2 = 0, or 2 + 2 = The Cat's Meow. There's no law of nature saying you can't. People make all this stuff up. Sure, what math people make up may prove useful, or fashionable, or impressive. And for that matter much of what math people invent can cause great consternation for some people on internet forums when some guy comes along and tries to tell them it's invented. But no matter how much they protest calling that revelation things like "a full-on, OCD-triggering horror show for any pure mathematician," it doesn't change the fact that math is neither absolute nor is it objective.

Discovered, not invented, or made up.

You asserted it. Now prove that assertion. You'll need to demonstrate that math exists prior to its discovery.

That "mathematics is discovered" is not a proposition. It's not something to be proven true or false. It's a heuristic intended to reveal the underlying philosophical commitments of pure mathematicians. So long as you look at mathematics as something that's created or invented, you're blinding yourself to how pure mathematicians approach their subject.

Mathematicians do create descriptions and invent methods, but that's not the mathematics itself. The math is the thing that was out there, self-existent before someone came along to notice it. It's the abstraction that exists even if no one ever discovers it. Math is the a priori that makes descriptions and methods possible.

Certainly, mathematics is fundamentally arbitrary. But to stop there misses the entirely objective restraints imposed by mathematics.

There are apocryphal tales that circulate among PhD candidates about a dissertation on a newly-discovered class of algebraic structures that had to be withdrawn when it was shown the class was empty. There's the actual tale of Principia Mathematica, the attempt by Russell and Whitehead to create a complete set-theoretic foundation for mathematics destroyed by Gödel in his "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I."

More prosaically, math is the essential that remains true even when mathematical models show that bumblebees can't fly.

I'd say that math is essentially invented. Laplace Transforms were invented by Laplace, and the Cartesian Coordinate System was invented by Descartes. We can see math being invented throughout the history of mathematics by many different people. All that math wasn't just lying around waiting to be dug up. So the knowledge that math is invented is based in the practice of basing mathematics in axioms. Axioms are arbitrary rules that people make up. As such, they are the product of human ingenuity and creativity (i.e. inventions). I've proved the role of arbitrary axioms in mathematics in my What proof is there that 2 + 2 = 4? thread.

That said, there is a discovery of sorts in math in which once some idea is invented, later on that idea leads to unforeseen conclusions. Circles, for example, were invented but later on the number π was found to be the ratio of the circumference of any circle to the measure of its diameter. Nobody including the inventors of circles expected π.

Kinda, but you're getting it all bass ackwards. Nobody invented a circle.

A circle is the locus of points equidistant from a chosen center. It's not something that exists more than approximately in the real world. It didn't come into existence the first time some primate rounded off a rock to help it roll better. It always existed. And even that's too restrictive. Its existence lies outside the bounds of space and time, independent of any potential or conceivable universe.

Neither is the ratio between circumference and diameter something that can be measured.

That's the math you're missing by insisting that math depends on its applications. This is the divide between pure and applied mathematics.

The descriptions codified by Laplace, and later Heaviside, and Descartes are indeed their own inventions. But the relationships described by Laplace and Descartes must have existed before they created those descriptions or there'd be nothing for them to describe. And those relationships would still exist if Laplace or Descartes were never born.

Yes, just lying around waiting to be dug up.

And again, mathematicians don't care if the mathematics they discover is useful.

That's flat-out false. Newton, for example, did some major work in developing calculus because he needed it to do his work in physics.

Perhaps not the best example. There's a reason we use Leibniz notation, and no committee secretly chaired by Newton granting him priority is going to change that.

In the sense that even the purest of mathematicians has to eat, yes, we care if somebody's willing to pay us to pursue our research. But pure math is abstract art for geeks. It's paradoxically described as the search for transuniversal truths that some physicist or other scientist won't someday sully with an application.

The search is hopeless, by the way. When algebraic number theory was found to have applications in cybersecurity, for many of us, the last of our dreams were crushed. George Boole had a great run, by the way.

As it turns out, surprisingly enough, it is, but that's irrelevant to the underlying abstractions that we work with.

That's a fact.

Certainly it's a fact that math isn't objective and is no more absolute than the arbitrarily accepted axioms upon which theory is based.

Right! So you get it. Why are you arguing with me?

Because your proofs are nonsense.

But it's not because two can't be represented in binary...

The conventional representation of 2 in binary is 10. Somebody cooked that up, of course.

...or because the residues in modular arithmetic aren't equivalence classes or because of any other contrafactual you've somehow absorbed and can't seem to keep yourself from sharing.

I do like to share my knowledge, that's true.

And you also share your beliefs that a representations of the number two using positional notation is the number itself. Which is nonsense.

It's one thing to claim something is true. It's another thing to support that claim. Congratulations on the former.

But I do support my claims. By contrast, your earlier claim that math is discovered is completely unsupported. The kettle calls the pot black!

If only you'd stopped when you were ahead.

I can understand why you want me to stop.

Ice cream has no bones. That's a true statement that supports no theorem in mathematics. The digit 2 doesn't exist in binary representations. Neither is it true that 2 + 2 isn't 4 because the digit 2 doesn't exist in binary representations.

Would you like to see a proof?

\(10_b+10_b=100_b\)

As for me, I can live with invented, subjective, and relative math. It's still elegant, fun, and very challenging. Heck, I study it hours a day--every day. And I've been learning it in school and through self-study for decades. After all that hard work, I can go online to see what others think of it. And no surprise--I see they've made it into an idol.

Read all the books you like. I commend autodidactism.

I'll keep reading books. I recommend you read some books too.

What is the point of comments like this?

But no, that's not you.

What's not I?

You're not an audidact.

There's no benefit in reading a book you can't understand, and less than no benefit when your lack of comprehension causes you to learn things that are not true.

I'll let you know if that ever happens!

Only if you discover it, and probably not then, either.

But honestly, many of the books I study can be hard for me to understand. But I would be an idiot to take your advice and stop studying them for that reason! As I see it, if I study ten new concepts, and I only understand one of those concepts, then I've learned one concept.

First of all, no, that's not what I'm saying. Yes, read books. But once you've read them, engage in the critical examination necessary to ensure comprehension. The former without the latter leads to the garbled miscomprehensions evident in your posts here.

If the book is a math or science text, it will have exercises listed after every topic. And answers to even problems listed in an appendix, generally, for the benefit of informal students like yourself.

Use the questions to critically examine whether you've actually learned the concepts. Or ask someone who's already mastered the concept to go over it with you. Or ask the professor who wrote the book. You'd be amazed how open they are to responding to correspondence.

Hell, I've got correspondence from Neil deGrasse Tyson in my email. Because I wrote him out of the blue with no introduction, and he wrote me back.

In any case, self-study has been very beneficial for me. When I was in college I prepared for many of my courses by studying beforehand. I ended up with a four-year degree and a 4.0 GPA.

The scary thing is that that's actually possible.

That's the time to seek help from reliable sources who can steer you away from some of the bizarre claims you've been making.

You're making one of the biggest goofs in mathematics here: You are relying on intuition and rejecting whatever seems strange to you. Many truths in mathematics as well as science are often counterintuitive. Truth doesn't care if it makes sense to us.

And those "bizarre claims" you mention are all based in conventional mathematics and logic. I'm not making up anything.

And it's also possible that your miscomprehensions are not original.

That's not personal. It's something that happens to all of us. I had a whole chapter of my dissertation wiped because one member of my committee noticed I'd headed off on a bridge to nowhere. The point being that without independent critical examination, there are no guard rails, and without guard rails, running off the cliff is a question of when, not if.

I can only wonder what they would think of your latest post.

That I'm wasting my time, probably.

I've got plenty of other students to keep me busy.

Send them my way. I've worked as a math tutor. They may well need one.

Yes, they do need tutors, often enough, but no, that's not you, either. We have a budget for tutors. The applicants must at minimum be enrolled in a graduate program.

A great video to watch that explains my position on mathematics is Philosophical Failures of Christian Apologetics, Part 4: Word Games. Note that Christian apologists see mathematics in a way that is similar to the way you see it.

Because closing with insults is just how you roll, right?

Your position on mathematics lacks the relevance of a mathematician's position on mathematics. Something you can only learn by asking mathematicians about their positions. And something you will never learn so long as you strive to impose your own limitations on others.

Where are your solutions to the questions you posed in the o/p?