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"Veritasium" (YT) - "The Big Misconception About Electricity" ?
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snarkysparky:
The fundamental theorem of algebra says

Any polynomial of degree N will have N roots or solutions.

You can plot Y = x^2 +1 

It never crosses the x axis.  Yet the FTA says it has two roots.

The FTA has many proofs as I gather from some googling. 

See if you can invalidate one of the proofs.

https://mathbitsnotebook.com/Algebra2/Polynomials/POfundamentalThm.html

Here is a proof to get you started.

https://www.math.ucdavis.edu/~anne/WQ2007/mat67-Ld-FTA.pdf

penfold:

--- Quote from: bsfeechannel on April 24, 2022, 04:06:58 pm ---
--- Quote from: adx on April 21, 2022, 04:42:51 am ---Being expected to trust in intellectual authorities with absolutely nil room for deviation is pretty much the definition of faith. All this talk of theorems, deductions and postulates is the setting up of a system to engender belief.
--- End quote ---
There is plenty of room for "deviation". The thing is that no one has been able, as of this day, to come up with something better.
[...]

--- End quote ---

But... whilst it is faith and trust, the faith and trust should be in the rationalism and logical framework in which maths exists, as with science where it is the scientific method in which we must trust and believe - it is just a necesary contradiction that we must trust prior work to be valid, though proof and review processes contribute to rationalising that assumption. So, surely the argument there is that the maner in which maths is presented to engineering students, in the non-rigorous sense (i.e. very different to maths-degree maths), the student is expected to assume what is presented as true... but must trust the logic from which it is derived.


--- Quote from: snarkysparky on April 24, 2022, 09:54:25 pm ---The fundamental theorem of algebra says

Any polynomial of degree N will have N roots or solutions.
[...]

--- End quote ---

But... that is a theorem of algebra, the complex number does not arrive until one starts to pose questions. Starting with natural numbers, all positive, whole numbered, countable, possesable etc quantities, we seek the answer to a+b=1 which is not defined for all both "a" and "b" in the set of natural numbers, enter the integer, the rational, the irrational and complex as we seek more or less general solutions to problems involving numbers in each set. But that's all find and dandy, but it isn't a general property of all sets of numbers and any relationship with reality depends on the formulation of the problem and it is a later attribution of significance which gives the numbers and significance or relationship to reality.
SandyCox:

--- Quote from: penfold on April 25, 2022, 08:27:37 am ---
--- Quote from: bsfeechannel on April 24, 2022, 04:06:58 pm ---
--- Quote from: adx on April 21, 2022, 04:42:51 am ---Being expected to trust in intellectual authorities with absolutely nil room for deviation is pretty much the definition of faith. All this talk of theorems, deductions and postulates is the setting up of a system to engender belief.
--- End quote ---
There is plenty of room for "deviation". The thing is that no one has been able, as of this day, to come up with something better.
[...]

--- End quote ---

But... whilst it is faith and trust, the faith and trust should be in the rationalism and logical framework in which maths exists, as with science where it is the scientific method in which we must trust and believe - it is just a necesary contradiction that we must trust prior work to be valid, though proof and review processes contribute to rationalising that assumption. So, surely the argument there is that the maner in which maths is presented to engineering students, in the non-rigorous sense (i.e. very different to maths-degree maths), the student is expected to assume what is presented as true... but must trust the logic from which it is derived.


--- Quote from: snarkysparky on April 24, 2022, 09:54:25 pm ---The fundamental theorem of algebra says

Any polynomial of degree N will have N roots or solutions.
[...]

--- End quote ---

But... that is a theorem of algebra, the complex number does not arrive until one starts to pose questions. Starting with natural numbers, all positive, whole numbered, countable, possesable etc quantities, we seek the answer to a+b=1 which is not defined for all both "a" and "b" in the set of natural numbers, enter the integer, the rational, the irrational and complex as we seek more or less general solutions to problems involving numbers in each set. But that's all find and dandy, but it isn't a general property of all sets of numbers and any relationship with reality depends on the formulation of the problem and it is a later attribution of significance which gives the numbers and significance or relationship to reality.

--- End quote ---

You should read more carefully.

The fundamental theorem of algebra states that any polynomial of degree N will have N roots over the Complex numbers.
penfold:

--- Quote from: SandyCox on April 25, 2022, 03:51:48 pm ---
--- Quote from: penfold on April 25, 2022, 08:27:37 am ---[...]

--- Quote from: snarkysparky on April 24, 2022, 09:54:25 pm ---The fundamental theorem of algebra says

Any polynomial of degree N will have N roots or solutions.
[...]

--- End quote ---

But... that is a theorem of algebra, the complex number does not arrive until one starts to pose questions. Starting with natural numbers, all positive, whole numbered, countable, possesable etc quantities, we seek the answer to a+b=1 which is not defined for all both "a" and "b" in the set of natural numbers, enter the integer, the rational, the irrational and complex as we seek more or less general solutions to problems involving numbers in each set. But that's all find and dandy, but it isn't a general property of all sets of numbers and any relationship with reality depends on the formulation of the problem and it is a later attribution of significance which gives the numbers and significance or relationship to reality.

--- End quote ---

You should read more carefully.

The fundamental theorem of algebra states that any polynomial of degree N will have N roots over the Complex numbers.

--- End quote ---

Had I stated otherwise? Or did you pick up on my deviation to set theory? In either case, the relationship between, say, the width of a square field and the number of square cars I can tessellate within it does not intrinsically result in any complex numbers until I pose the question of what width would I need to hold a negative number of cars... my point remaining that the definition of the problem and how the problem is abstracted is important to prevent the descartian absurdity of "imaginary" numbers which the abstract nature of modern maths avoids by removing that very intrinsic link between numbers on the page and measureable quantities in reality.
bsfeechannel:

--- Quote from: penfold on April 25, 2022, 08:27:37 am ---But... whilst it is faith and trust, the faith and trust should be in the rationalism and logical framework in which maths exists, as with science where it is the scientific method in which we must trust and believe - it is just a necesary contradiction that we must trust prior work to be valid, though proof and review processes contribute to rationalising that assumption.

--- End quote ---

We don't trust science. Science is just a method for accumulating knowledge based exactly on distrusting current hypotheses.


--- Quote --- So, surely the argument there is that the maner in which maths is presented to engineering students, in the non-rigorous sense (i.e. very different to maths-degree maths), the student is expected to assume what is presented as true... but must trust the logic from which it is derived.
--- End quote ---

I don't know where you had your engineering math courses, and I don't care, but where I learned about math, still in high school, they taught us that math has axioms, or postulates, that are provisional truths, subject to denial if convenient.

It is the case for instance of the so called parallel postulate: true in euclidean geometry; false in, you guessed it, non-euclidean geometry (that one Einstein used for the GTR). 

When we arrived in college, for our engineering degree, we all had this concept in mind. Postulates were accepted as ad hoc truths, we had to prove the deductions from these postulates and then test their application in the lab.

No one told us to trust or believe anything.

If the experience you had with math in your engineering degree is the one you described, I feel bad for you.
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