#debunking — Public Fediverse posts

Here we are again on #MathsMonday, looking at the questionable statements of SmartmanApps, #debunking the #disinformation and seeing what real #mathematics it accidentally uncovers.

Our now-familiar friend says that infinitesimals are a standard thing taught in schools. This is false, with the possible exception of wherever he teaches, because infinitesimals have by and large been ejected from mathematics due to being unnecessary and confusing.

But what is true is that infinitesimals *used* to be taught, and he has the very old textbooks to prove it. Let’s find one of those textbooks, and its definition of infinitesimal:

> lim a = 0 means that a is an infinitesimal quantity

(Advanced Algebra, p226 by Collins, J. 1911. American Book Company. Symbols altered for typographical necessity)

So we need to know what “lim x = a” means, which is furnished to us on page 297 (it is no credit to this textbook that it uses undefined terms without even an apology):

> If a variable x assumes a given sequence of values such that the numerical difference between a constant a, and the variable x becomes and remains less than any assignable quantity, however small, then x is said to approach a as its limit [written] lim x = a

This is essentially the way Cauchy defined limits, and is the same as the definition of a limit I gave last time, with two changes:

1. It’s less easily translated into the symbolic version due to not using directly equivalent language like “for every”.
2. Instead of a sequence, we talk of a “variable”.

It is the second difference which is more consequential. In modern mathematics, in any given context, a variable only takes on a single value. If you think of x as possibly taking on multiple values, say 2 and 0 in alternation, then the following argument no longer works:

“x is not less than 1, and x is not greater than 1, therefore x is equal to 1”.

Imagine asking “is x less than 1?” and finding that x could be 2, so answering, “no”; then we may ask “is x greater than 1?” and find that x could be 0, so answer “no” again. But concluding that x is equal to 1 is not valid. So, we must be smarter about such arguments. Maybe our Smart friend is that smart, but I would prefer not to depend on it.

The solution modern mathematics has taken to this is that if we want to consider a changing quantity, we encode it into a different kind of object: a function. The function accepts a parameter, and that parameter determines what value it returns. Thus the dependence of that value on the parameter is made explicit; we save ourselves from the possibility of getting confused, because “x(0) is not less than 1, and x(1) is not greater than 1, therefore x(2) is equal to 1” is obviously rubbish!

In school it is typical still to describe functions in the old-fashioned language of variables, so you probably remember seeing a linear function described by “y = mx + c”, where x is a variable, and y another variable which *depends* on it. Further on in school it’s typical to replace “y” with “f(x)” which makes this dependence explicit.

Because functions are an entirely different type of thing, we also never compare them to numbers without first describing exactly how that comparison should happen. We don’t ask, “is f equal to 1”? Because the answer is clearly “no”, and similarly f is not less than or greater than 1; f is a function, not a number so these questions don’t even really make sense. This is not the case with variables; it is perfectly reasonable to ask whether x is equal to 1 - this kind of question is embedded in every single equation a school pupil is asked to solve.

Clearly, mathematicians and pupils of yesteryear were all able to work like this, but it is unnecessarily confusing and error-prone.

## Sequences

So, we will use functions instead of variables, and since it will not cause any extra difficulties, we will specifically use sequences which are functions whose inputs are natural numbers. This turns Cauchy’s definition of the limit into the modern one, and it turns his definition of an infinitesimal into “a sequence with a limit of zero”.

We have some questions to answer: while it is natural to compare a variable to numbers (such as when solving a system of inequalities) comparing a function (or sequence) to a number is less natural. What does “sin < ½” mean? Is it true, or false? Clearly sin(x) < ½ for some x but not for others.

Nonetheless, if we want sequences to occupy the role of infinitesimals, we need to be able to perform such comparisons, because the fundamental property of an infinitesimal e is that 0 < e < 1/n for all natural numbers n. Thus how we set up infinitesimals is fundamentally a matter of how we set up rules for comparing the sequences which represent them.

Let us first point out that instead of comparing sequences to *numbers* we can replace any fixed number with the constant sequence all of whose elements are that fixed number. And secondly, let’s establish that the *normal* equality of sequences is simply that each of their corresponding elements must be equal, that is, their first elements must be equal to each other, their second elements must be equal to each other, and so on. This means that there are many sequences whose limit is zero but which are not equal to one another. One of those sequences is the constant sequence all of whose elements are zero, i.e. the sequence which we are using instead of the number zero, written (0, 0, 0, …)

I should point out here that it is a standard fact about limits that the limit of a constant sequence is always that constant. This should be easy to see from the definition, but in a university course you would prove it rigorously. This means, also, that any sequence whose limit is zero must be different from any constant sequence, except that the zero sequence is equal to itself. This is good for our use of sequences which converge to zero (i.e. have limit zero) as infinitesimals, since they should be different from numbers larger than zero.

## Equivalence and Order

But we still have to do more work, because to be infinitesimals, these sequences need to be smaller than 1/n, for every natural number n. We know the first step to making sense of this: replacing 1/n with the sequence (1/n, 1/n, 1/n, …). The next step is creating a notion of “smaller” and “larger” sequences.

In what follows, I will fix a sequence e representing an infinitesimal, e := (1, ½, ⅓, ¼, …), and x representing a small finite positive number, x:= (0.1, 0.1, 0.1, …)

### Global Domination

One very simple way of setting up an ordering is to say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1 and a2 < b2 and a3 < b3, and so on, for each index. This is no good for our purpose, because look at e and x. If we are doing things right, we should have e < x, but e1 = 1 is not less than x1 = 0.1. Even worse, if we go to the 11th element, x11 = 0.1 is not less than e11 = 1/11, so neither e < x nor x < e: this is not a **total** ordering, i.e. sometimes two objects just do not lie in any particular order.

### Lexicographic Ordering

The lexicographic (so called because it’s the ordering used by dictionaries - i.e. lexicons) ordering is perhaps the next most obvious way of ordering something that’s made out of multiple things which are themselves ordered.

Using this ordering we’d say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1, or a1 = b1 and a2 < b2, or a1 = b1 and a2 = b2 and a3 < b3, or … and so on.

This *does* give us a total order; we can compare any two sequences and either one is less than the other, or they are exactly the same sequence. However, we still have a problem because we find that since 0.1 < 1, we have defined x < e, which is not what we wanted.

This ordering fails to capture the idea that it is the *eventual* behaviour of infinitesimals that we care about, and instead makes the early behaviour the most important.

### Eventual Domination

We can’t just do a “reverse lexicographic order” because we can’t start at the *last* element of a sequence composed of infinitely many elements.[^4] But we can do something else: we can say that (a1, a2, a3, …) ≤ (b1, b2, b3, …) if *from some point* all the a’s are smaller than all the b’s. You can hopefully see that, with this ordering, e ≤ x: from the 11th element onwards, all e’s are below 0.1. And indeed if y is *any* constant sequence consisting of rational numbers, eventually e will drop below that number and stay below.

It is not for no reason that I used ≤ in this definition instead of <. That is because this is again not a total order among all possible sequences: a sequence could oscillate above and below another sequence forever.

### Ordinary Real Numbers

What Cantor did in his development of real analysis was to notice that there is a certain kind of sequence that behaves very nicely, which nowadays we call a *Cauchy sequence*. It has the property that as the sequence continues, its values become arbitrarily close to all later values:

> A sequence a is *Cauchy* if, for every ε > 0, there is a number N such that for all n, m both greater than N, |a(n) - a(m)| < ε

The structure Cantor worked with was all of *these* sequences (of rational numbers) rather than arbitrary sequences. Then, he said that two sequences were equivalent if the difference between them becomes arbitrarily small. If not, then (one can prove) the difference between the values of a and b is either eventually positive, in which case we say a is larger, or eventually negative and we say b is larger.

We can go on to define arithmetic between such sequences and prove that everything works out wonderfully but there is one snag for us: there are *no* infinitesimals! What was the fate of our supposedly infinitesimal sequence e? Well, unfortunately the difference between it and the zero sequences becomes arbitrarily small, so they are rendered equivalent: any sequence which is “infinitesimal” in the Cauchy sense becomes merely zero.

[^4]: Unless the infinite order type is a *successor ordinal* which is a concept beyond the scope of this post, and is not the case for our sequences.

#math #maths

1/2

Here we are again on #MathsMonday, looking at the questionable statements of SmartmanApps, #debunking the #disinformation and seeing what real #mathematics it accidentally uncovers.

Our now-familiar friend says that infinitesimals are a standard thing taught in schools. This is false, with the possible exception of wherever he teaches, because infinitesimals have by and large been ejected from mathematics due to being unnecessary and confusing.

But what is true is that infinitesimals *used* to be taught, and he has the very old textbooks to prove it. Let’s find one of those textbooks, and its definition of infinitesimal:

> lim a = 0 means that a is an infinitesimal quantity

(Advanced Algebra, p226 by Collins, J. 1911. American Book Company. Symbols altered for typographical necessity)

So we need to know what “lim x = a” means, which is furnished to us on page 297 (it is no credit to this textbook that it uses undefined terms without even an apology):

> If a variable x assumes a given sequence of values such that the numerical difference between a constant a, and the variable x becomes and remains less than any assignable quantity, however small, then x is said to approach a as its limit [written] lim x = a

This is essentially the way Cauchy defined limits, and is the same as the definition of a limit I gave last time, with two changes:

1. It’s less easily translated into the symbolic version due to not using directly equivalent language like “for every”.
2. Instead of a sequence, we talk of a “variable”.

It is the second difference which is more consequential. In modern mathematics, in any given context, a variable only takes on a single value. If you think of x as possibly taking on multiple values, say 2 and 0 in alternation, then the following argument no longer works:

“x is not less than 1, and x is not greater than 1, therefore x is equal to 1”.

Imagine asking “is x less than 1?” and finding that x could be 2, so answering, “no”; then we may ask “is x greater than 1?” and find that x could be 0, so answer “no” again. But concluding that x is equal to 1 is not valid. So, we must be smarter about such arguments. Maybe our Smart friend is that smart, but I would prefer not to depend on it.

The solution modern mathematics has taken to this is that if we want to consider a changing quantity, we encode it into a different kind of object: a function. The function accepts a parameter, and that parameter determines what value it returns. Thus the dependence of that value on the parameter is made explicit; we save ourselves from the possibility of getting confused, because “x(0) is not less than 1, and x(1) is not greater than 1, therefore x(2) is equal to 1” is obviously rubbish!

In school it is typical still to describe functions in the old-fashioned language of variables, so you probably remember seeing a linear function described by “y = mx + c”, where x is a variable, and y another variable which *depends* on it. Further on in school it’s typical to replace “y” with “f(x)” which makes this dependence explicit.

Because functions are an entirely different type of thing, we also never compare them to numbers without first describing exactly how that comparison should happen. We don’t ask, “is f equal to 1”? Because the answer is clearly “no”, and similarly f is not less than or greater than 1; f is a function, not a number so these questions don’t even really make sense. This is not the case with variables; it is perfectly reasonable to ask whether x is equal to 1 - this kind of question is embedded in every single equation a school pupil is asked to solve.

Clearly, mathematicians and pupils of yesteryear were all able to work like this, but it is unnecessarily confusing and error-prone.

## Sequences

So, we will use functions instead of variables, and since it will not cause any extra difficulties, we will specifically use sequences which are functions whose inputs are natural numbers. This turns Cauchy’s definition of the limit into the modern one, and it turns his definition of an infinitesimal into “a sequence with a limit of zero”.

We have some questions to answer: while it is natural to compare a variable to numbers (such as when solving a system of inequalities) comparing a function (or sequence) to a number is less natural. What does “sin < ½” mean? Is it true, or false? Clearly sin(x) < ½ for some x but not for others.

Nonetheless, if we want sequences to occupy the role of infinitesimals, we need to be able to perform such comparisons, because the fundamental property of an infinitesimal e is that 0 < e < 1/n for all natural numbers n. Thus how we set up infinitesimals is fundamentally a matter of how we set up rules for comparing the sequences which represent them.

Let us first point out that instead of comparing sequences to *numbers* we can replace any fixed number with the constant sequence all of whose elements are that fixed number. And secondly, let’s establish that the *normal* equality of sequences is simply that each of their corresponding elements must be equal, that is, their first elements must be equal to each other, their second elements must be equal to each other, and so on. This means that there are many sequences whose limit is zero but which are not equal to one another. One of those sequences is the constant sequence all of whose elements are zero, i.e. the sequence which we are using instead of the number zero, written (0, 0, 0, …)

I should point out here that it is a standard fact about limits that the limit of a constant sequence is always that constant. This should be easy to see from the definition, but in a university course you would prove it rigorously. This means, also, that any sequence whose limit is zero must be different from any constant sequence, except that the zero sequence is equal to itself. This is good for our use of sequences which converge to zero (i.e. have limit zero) as infinitesimals, since they should be different from numbers larger than zero.

## Equivalence and Order

But we still have to do more work, because to be infinitesimals, these sequences need to be smaller than 1/n, for every natural number n. We know the first step to making sense of this: replacing 1/n with the sequence (1/n, 1/n, 1/n, …). The next step is creating a notion of “smaller” and “larger” sequences.

In what follows, I will fix a sequence e representing an infinitesimal, e := (1, ½, ⅓, ¼, …), and x representing a small finite positive number, x:= (0.1, 0.1, 0.1, …)

### Global Domination

One very simple way of setting up an ordering is to say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1 and a2 < b2 and a3 < b3, and so on, for each index. This is no good for our purpose, because look at e and x. If we are doing things right, we should have e < x, but e1 = 1 is not less than x1 = 0.1. Even worse, if we go to the 11th element, x11 = 0.1 is not less than e11 = 1/11, so neither e < x nor x < e: this is not a **total** ordering, i.e. sometimes two objects just do not lie in any particular order.

### Lexicographic Ordering

The lexicographic (so called because it’s the ordering used by dictionaries - i.e. lexicons) ordering is perhaps the next most obvious way of ordering something that’s made out of multiple things which are themselves ordered.

Using this ordering we’d say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1, or a1 = b1 and a2 < b2, or a1 = b1 and a2 = b2 and a3 < b3, or … and so on.

This *does* give us a total order; we can compare any two sequences and either one is less than the other, or they are exactly the same sequence. However, we still have a problem because we find that since 0.1 < 1, we have defined x < e, which is not what we wanted.

This ordering fails to capture the idea that it is the *eventual* behaviour of infinitesimals that we care about, and instead makes the early behaviour the most important.

### Eventual Domination

We can’t just do a “reverse lexicographic order” because we can’t start at the *last* element of a sequence composed of infinitely many elements.[^4] But we can do something else: we can say that (a1, a2, a3, …) ≤ (b1, b2, b3, …) if *from some point* all the a’s are smaller than all the b’s. You can hopefully see that, with this ordering, e ≤ x: from the 11th element onwards, all e’s are below 0.1. And indeed if y is *any* constant sequence consisting of rational numbers, eventually e will drop below that number and stay below.

It is not for no reason that I used ≤ in this definition instead of <. That is because this is again not a total order among all possible sequences: a sequence could oscillate above and below another sequence forever.

### Ordinary Real Numbers

What Cantor did in his development of real analysis was to notice that there is a certain kind of sequence that behaves very nicely, which nowadays we call a *Cauchy sequence*. It has the property that as the sequence continues, its values become arbitrarily close to all later values:

> A sequence a is *Cauchy* if, for every ε > 0, there is a number N such that for all n, m both greater than N, |a(n) - a(m)| < ε

The structure Cantor worked with was all of *these* sequences (of rational numbers) rather than arbitrary sequences. Then, he said that two sequences were equivalent if the difference between them becomes arbitrarily small. If not, then (one can prove) the difference between the values of a and b is either eventually positive, in which case we say a is larger, or eventually negative and we say b is larger.

We can go on to define arithmetic between such sequences and prove that everything works out wonderfully but there is one snag for us: there are *no* infinitesimals! What was the fate of our supposedly infinitesimal sequence e? Well, unfortunately the difference between it and the zero sequences becomes arbitrarily small, so they are rendered equivalent: any sequence which is “infinitesimal” in the Cauchy sense becomes merely zero.

[^4]: Unless the infinite order type is a *successor ordinal* which is a concept beyond the scope of this post, and is not the case for our sequences.

#math #maths

1/2

Here we are again on #MathsMonday, looking at the questionable statements of SmartmanApps, #debunking the #disinformation and seeing what real #mathematics it accidentally uncovers.

Our now-familiar friend says that infinitesimals are a standard thing taught in schools. This is false, with the possible exception of wherever he teaches, because infinitesimals have by and large been ejected from mathematics due to being unnecessary and confusing.

But what is true is that infinitesimals *used* to be taught, and he has the very old textbooks to prove it. Let’s find one of those textbooks, and its definition of infinitesimal:

> lim a = 0 means that a is an infinitesimal quantity

(Advanced Algebra, p226 by Collins, J. 1911. American Book Company. Symbols altered for typographical necessity)

So we need to know what “lim x = a” means, which is furnished to us on page 297 (it is no credit to this textbook that it uses undefined terms without even an apology):

> If a variable x assumes a given sequence of values such that the numerical difference between a constant a, and the variable x becomes and remains less than any assignable quantity, however small, then x is said to approach a as its limit [written] lim x = a

This is essentially the way Cauchy defined limits, and is the same as the definition of a limit I gave last time, with two changes:

1. It’s less easily translated into the symbolic version due to not using directly equivalent language like “for every”.
2. Instead of a sequence, we talk of a “variable”.

It is the second difference which is more consequential. In modern mathematics, in any given context, a variable only takes on a single value. If you think of x as possibly taking on multiple values, say 2 and 0 in alternation, then the following argument no longer works:

“x is not less than 1, and x is not greater than 1, therefore x is equal to 1”.

Imagine asking “is x less than 1?” and finding that x could be 2, so answering, “no”; then we may ask “is x greater than 1?” and find that x could be 0, so answer “no” again. But concluding that x is equal to 1 is not valid. So, we must be smarter about such arguments. Maybe our Smart friend is that smart, but I would prefer not to depend on it.

The solution modern mathematics has taken to this is that if we want to consider a changing quantity, we encode it into a different kind of object: a function. The function accepts a parameter, and that parameter determines what value it returns. Thus the dependence of that value on the parameter is made explicit; we save ourselves from the possibility of getting confused, because “x(0) is not less than 1, and x(1) is not greater than 1, therefore x(2) is equal to 1” is obviously rubbish!

In school it is typical still to describe functions in the old-fashioned language of variables, so you probably remember seeing a linear function described by “y = mx + c”, where x is a variable, and y another variable which *depends* on it. Further on in school it’s typical to replace “y” with “f(x)” which makes this dependence explicit.

Because functions are an entirely different type of thing, we also never compare them to numbers without first describing exactly how that comparison should happen. We don’t ask, “is f equal to 1”? Because the answer is clearly “no”, and similarly f is not less than or greater than 1; f is a function, not a number so these questions don’t even really make sense. This is not the case with variables; it is perfectly reasonable to ask whether x is equal to 1 - this kind of question is embedded in every single equation a school pupil is asked to solve.

Clearly, mathematicians and pupils of yesteryear were all able to work like this, but it is unnecessarily confusing and error-prone.

## Sequences

So, we will use functions instead of variables, and since it will not cause any extra difficulties, we will specifically use sequences which are functions whose inputs are natural numbers. This turns Cauchy’s definition of the limit into the modern one, and it turns his definition of an infinitesimal into “a sequence with a limit of zero”.

We have some questions to answer: while it is natural to compare a variable to numbers (such as when solving a system of inequalities) comparing a function (or sequence) to a number is less natural. What does “sin < ½” mean? Is it true, or false? Clearly sin(x) < ½ for some x but not for others.

Nonetheless, if we want sequences to occupy the role of infinitesimals, we need to be able to perform such comparisons, because the fundamental property of an infinitesimal e is that 0 < e < 1/n for all natural numbers n. Thus how we set up infinitesimals is fundamentally a matter of how we set up rules for comparing the sequences which represent them.

Let us first point out that instead of comparing sequences to *numbers* we can replace any fixed number with the constant sequence all of whose elements are that fixed number. And secondly, let’s establish that the *normal* equality of sequences is simply that each of their corresponding elements must be equal, that is, their first elements must be equal to each other, their second elements must be equal to each other, and so on. This means that there are many sequences whose limit is zero but which are not equal to one another. One of those sequences is the constant sequence all of whose elements are zero, i.e. the sequence which we are using instead of the number zero, written (0, 0, 0, …)

I should point out here that it is a standard fact about limits that the limit of a constant sequence is always that constant. This should be easy to see from the definition, but in a university course you would prove it rigorously. This means, also, that any sequence whose limit is zero must be different from any constant sequence, except that the zero sequence is equal to itself. This is good for our use of sequences which converge to zero (i.e. have limit zero) as infinitesimals, since they should be different from numbers larger than zero.

## Equivalence and Order

But we still have to do more work, because to be infinitesimals, these sequences need to be smaller than 1/n, for every natural number n. We know the first step to making sense of this: replacing 1/n with the sequence (1/n, 1/n, 1/n, …). The next step is creating a notion of “smaller” and “larger” sequences.

In what follows, I will fix a sequence e representing an infinitesimal, e := (1, ½, ⅓, ¼, …), and x representing a small finite positive number, x:= (0.1, 0.1, 0.1, …)

### Global Domination

One very simple way of setting up an ordering is to say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1 and a2 < b2 and a3 < b3, and so on, for each index. This is no good for our purpose, because look at e and x. If we are doing things right, we should have e < x, but e1 = 1 is not less than x1 = 0.1. Even worse, if we go to the 11th element, x11 = 0.1 is not less than e11 = 1/11, so neither e < x nor x < e: this is not a **total** ordering, i.e. sometimes two objects just do not lie in any particular order.

### Lexicographic Ordering

The lexicographic (so called because it’s the ordering used by dictionaries - i.e. lexicons) ordering is perhaps the next most obvious way of ordering something that’s made out of multiple things which are themselves ordered.

Using this ordering we’d say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1, or a1 = b1 and a2 < b2, or a1 = b1 and a2 = b2 and a3 < b3, or … and so on.

This *does* give us a total order; we can compare any two sequences and either one is less than the other, or they are exactly the same sequence. However, we still have a problem because we find that since 0.1 < 1, we have defined x < e, which is not what we wanted.

This ordering fails to capture the idea that it is the *eventual* behaviour of infinitesimals that we care about, and instead makes the early behaviour the most important.

### Eventual Domination

We can’t just do a “reverse lexicographic order” because we can’t start at the *last* element of a sequence composed of infinitely many elements.[^4] But we can do something else: we can say that (a1, a2, a3, …) ≤ (b1, b2, b3, …) if *from some point* all the a’s are smaller than all the b’s. You can hopefully see that, with this ordering, e ≤ x: from the 11th element onwards, all e’s are below 0.1. And indeed if y is *any* constant sequence consisting of rational numbers, eventually e will drop below that number and stay below.

It is not for no reason that I used ≤ in this definition instead of <. That is because this is again not a total order among all possible sequences: a sequence could oscillate above and below another sequence forever.

### Ordinary Real Numbers

What Cantor did in his development of real analysis was to notice that there is a certain kind of sequence that behaves very nicely, which nowadays we call a *Cauchy sequence*. It has the property that as the sequence continues, its values become arbitrarily close to all later values:

> A sequence a is *Cauchy* if, for every ε > 0, there is a number N such that for all n, m both greater than N, |a(n) - a(m)| < ε

The structure Cantor worked with was all of *these* sequences (of rational numbers) rather than arbitrary sequences. Then, he said that two sequences were equivalent if the difference between them becomes arbitrarily small. If not, then (one can prove) the difference between the values of a and b is either eventually positive, in which case we say a is larger, or eventually negative and we say b is larger.

We can go on to define arithmetic between such sequences and prove that everything works out wonderfully but there is one snag for us: there are *no* infinitesimals! What was the fate of our supposedly infinitesimal sequence e? Well, unfortunately the difference between it and the zero sequences becomes arbitrarily small, so they are rendered equivalent: any sequence which is “infinitesimal” in the Cauchy sense becomes merely zero.

[^4]: Unless the infinite order type is a *successor ordinal* which is a concept beyond the scope of this post, and is not the case for our sequences.

#math #maths

1/2

Here we are again on #MathsMonday, looking at the questionable statements of SmartmanApps, #debunking the #disinformation and seeing what real #mathematics it accidentally uncovers.

Our now-familiar friend says that infinitesimals are a standard thing taught in schools. This is false, with the possible exception of wherever he teaches, because infinitesimals have by and large been ejected from mathematics due to being unnecessary and confusing.

But what is true is that infinitesimals *used* to be taught, and he has the very old textbooks to prove it. Let’s find one of those textbooks, and its definition of infinitesimal:

> lim a = 0 means that a is an infinitesimal quantity

(Advanced Algebra, p226 by Collins, J. 1911. American Book Company. Symbols altered for typographical necessity)

So we need to know what “lim x = a” means, which is furnished to us on page 297 (it is no credit to this textbook that it uses undefined terms without even an apology):

> If a variable x assumes a given sequence of values such that the numerical difference between a constant a, and the variable x becomes and remains less than any assignable quantity, however small, then x is said to approach a as its limit [written] lim x = a

This is essentially the way Cauchy defined limits, and is the same as the definition of a limit I gave last time, with two changes:

1. It’s less easily translated into the symbolic version due to not using directly equivalent language like “for every”.
2. Instead of a sequence, we talk of a “variable”.

It is the second difference which is more consequential. In modern mathematics, in any given context, a variable only takes on a single value. If you think of x as possibly taking on multiple values, say 2 and 0 in alternation, then the following argument no longer works:

“x is not less than 1, and x is not greater than 1, therefore x is equal to 1”.

Imagine asking “is x less than 1?” and finding that x could be 2, so answering, “no”; then we may ask “is x greater than 1?” and find that x could be 0, so answer “no” again. But concluding that x is equal to 1 is not valid. So, we must be smarter about such arguments. Maybe our Smart friend is that smart, but I would prefer not to depend on it.

The solution modern mathematics has taken to this is that if we want to consider a changing quantity, we encode it into a different kind of object: a function. The function accepts a parameter, and that parameter determines what value it returns. Thus the dependence of that value on the parameter is made explicit; we save ourselves from the possibility of getting confused, because “x(0) is not less than 1, and x(1) is not greater than 1, therefore x(2) is equal to 1” is obviously rubbish!

In school it is typical still to describe functions in the old-fashioned language of variables, so you probably remember seeing a linear function described by “y = mx + c”, where x is a variable, and y another variable which *depends* on it. Further on in school it’s typical to replace “y” with “f(x)” which makes this dependence explicit.

Because functions are an entirely different type of thing, we also never compare them to numbers without first describing exactly how that comparison should happen. We don’t ask, “is f equal to 1”? Because the answer is clearly “no”, and similarly f is not less than or greater than 1; f is a function, not a number so these questions don’t even really make sense. This is not the case with variables; it is perfectly reasonable to ask whether x is equal to 1 - this kind of question is embedded in every single equation a school pupil is asked to solve.

Clearly, mathematicians and pupils of yesteryear were all able to work like this, but it is unnecessarily confusing and error-prone.

## Sequences

So, we will use functions instead of variables, and since it will not cause any extra difficulties, we will specifically use sequences which are functions whose inputs are natural numbers. This turns Cauchy’s definition of the limit into the modern one, and it turns his definition of an infinitesimal into “a sequence with a limit of zero”.

We have some questions to answer: while it is natural to compare a variable to numbers (such as when solving a system of inequalities) comparing a function (or sequence) to a number is less natural. What does “sin < ½” mean? Is it true, or false? Clearly sin(x) < ½ for some x but not for others.

Nonetheless, if we want sequences to occupy the role of infinitesimals, we need to be able to perform such comparisons, because the fundamental property of an infinitesimal e is that 0 < e < 1/n for all natural numbers n. Thus how we set up infinitesimals is fundamentally a matter of how we set up rules for comparing the sequences which represent them.

Let us first point out that instead of comparing sequences to *numbers* we can replace any fixed number with the constant sequence all of whose elements are that fixed number. And secondly, let’s establish that the *normal* equality of sequences is simply that each of their corresponding elements must be equal, that is, their first elements must be equal to each other, their second elements must be equal to each other, and so on. This means that there are many sequences whose limit is zero but which are not equal to one another. One of those sequences is the constant sequence all of whose elements are zero, i.e. the sequence which we are using instead of the number zero, written (0, 0, 0, …)

I should point out here that it is a standard fact about limits that the limit of a constant sequence is always that constant. This should be easy to see from the definition, but in a university course you would prove it rigorously. This means, also, that any sequence whose limit is zero must be different from any constant sequence, except that the zero sequence is equal to itself. This is good for our use of sequences which converge to zero (i.e. have limit zero) as infinitesimals, since they should be different from numbers larger than zero.

## Equivalence and Order

But we still have to do more work, because to be infinitesimals, these sequences need to be smaller than 1/n, for every natural number n. We know the first step to making sense of this: replacing 1/n with the sequence (1/n, 1/n, 1/n, …). The next step is creating a notion of “smaller” and “larger” sequences.

In what follows, I will fix a sequence e representing an infinitesimal, e := (1, ½, ⅓, ¼, …), and x representing a small finite positive number, x:= (0.1, 0.1, 0.1, …)

### Global Domination

One very simple way of setting up an ordering is to say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1 and a2 < b2 and a3 < b3, and so on, for each index. This is no good for our purpose, because look at e and x. If we are doing things right, we should have e < x, but e1 = 1 is not less than x1 = 0.1. Even worse, if we go to the 11th element, x11 = 0.1 is not less than e11 = 1/11, so neither e < x nor x < e: this is not a **total** ordering, i.e. sometimes two objects just do not lie in any particular order.

### Lexicographic Ordering

The lexicographic (so called because it’s the ordering used by dictionaries - i.e. lexicons) ordering is perhaps the next most obvious way of ordering something that’s made out of multiple things which are themselves ordered.

Using this ordering we’d say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1, or a1 = b1 and a2 < b2, or a1 = b1 and a2 = b2 and a3 < b3, or … and so on.

This *does* give us a total order; we can compare any two sequences and either one is less than the other, or they are exactly the same sequence. However, we still have a problem because we find that since 0.1 < 1, we have defined x < e, which is not what we wanted.

This ordering fails to capture the idea that it is the *eventual* behaviour of infinitesimals that we care about, and instead makes the early behaviour the most important.

### Eventual Domination

We can’t just do a “reverse lexicographic order” because we can’t start at the *last* element of a sequence composed of infinitely many elements.[^4] But we can do something else: we can say that (a1, a2, a3, …) ≤ (b1, b2, b3, …) if *from some point* all the a’s are smaller than all the b’s. You can hopefully see that, with this ordering, e ≤ x: from the 11th element onwards, all e’s are below 0.1. And indeed if y is *any* constant sequence consisting of rational numbers, eventually e will drop below that number and stay below.

It is not for no reason that I used ≤ in this definition instead of <. That is because this is again not a total order among all possible sequences: a sequence could oscillate above and below another sequence forever.

### Ordinary Real Numbers

What Cantor did in his development of real analysis was to notice that there is a certain kind of sequence that behaves very nicely, which nowadays we call a *Cauchy sequence*. It has the property that as the sequence continues, its values become arbitrarily close to all later values:

> A sequence a is *Cauchy* if, for every ε > 0, there is a number N such that for all n, m both greater than N, |a(n) - a(m)| < ε

The structure Cantor worked with was all of *these* sequences (of rational numbers) rather than arbitrary sequences. Then, he said that two sequences were equivalent if the difference between them becomes arbitrarily small. If not, then (one can prove) the difference between the values of a and b is either eventually positive, in which case we say a is larger, or eventually negative and we say b is larger.

We can go on to define arithmetic between such sequences and prove that everything works out wonderfully but there is one snag for us: there are *no* infinitesimals! What was the fate of our supposedly infinitesimal sequence e? Well, unfortunately the difference between it and the zero sequences becomes arbitrarily small, so they are rendered equivalent: any sequence which is “infinitesimal” in the Cauchy sense becomes merely zero.

[^4]: Unless the infinite order type is a *successor ordinal* which is a concept beyond the scope of this post, and is not the case for our sequences.

#math #maths

1/2

https://www.europesays.com/uk/970057/ A grieving elephant, a dead Kruger ranger, and the AI slop that fooled social media #Animals #Conservation #debunking #KrugerNationalPark #Science #SocialMedia #UK #UnitedKingdom #wildlife

Time for another round of examining the interesting beliefs of SmartmanApps and #debunking the #disinformation this #MathsMonday.

We saw last time that his view of mathematics is at odds with that of the mainstream: I enumerated the standard axioms of the real numbers and proved that there can be no number 0.999… that is simultaneously less than 1 and greater than 0.9, 0.99, and any such finite decimal truncation of 0.999….

His idea is that 1 is “the limit of” 0.999…, but not the exact value of it. But what exactly is a limit? Let’s have a look at what our Smart Friend calls a limit:

> The limit is the number which [the sequence] never reaches

(see https://dotnet.social/@SmartmanApps/116303201093245275 I am not quite certain he intends this language to apply to all sequences, but I have not seen other descriptions from him)

This is simply inadequate, and it’s worth seeing why such poor explanations are inadequate with some examples.

* The sequence (1, 1, 1, …) *never reaches* the number 2, so is 2 the limit? The same applies for every number greater than 1, so are all of them the limit? The wording “the number” implies that the limit should be unique.
* On the other hand, it *does* reach 1, which intuition says ought to be the limit of this sequence.
* The sequence (1, 2, 3, 4, …) will exceed every number eventually and so, I guess, “reaches” every number. Again the wording “the number” suggests that such a number always exists, but apparently does not.

It may be that the Genius has a more precise idea of limit lurking in his mind, but to tease it out we’d have to interrogate him about these (and probably other) examples, and most likely anyone who tried would get blocked before they could complete their investigation.

The usual definition can be seen clearly from the early 19th century, due to Bolzano, though its roots go back further. That definition is:

> For a sequence (a_0, a_1, a_2, …), and a real number A, if for every real number ε > 0, there is some natural number N such that for every n > N, |a_n - A| < ε, then we say that A is the limit of the sequence (a_0, a_1, a_2, …).

This is quite a mouthful, and first year mathematics undergraduates spend quite some time getting the hang of it. A characteristic of the definition is the alternating *quantifiers*, which are written out “for every” and “there is” here (but would normally be written with symbols). It took mathematicians some time to come up with this modern version of quantified mathematical language.

Nevertheless we can put it into simpler language, at the cost of a little precision: **the limit of a sequence is A if the sequence gets as close to A as we like and remains that close forever**. It’s important that we keep that “remains that close” in. It’s important that neither of these ways of describing the concept assume a limit exists, because not all sequences have a limit. It is quite easy to prove directly from the definition and the properties of the real numbers that:

* A constant sequence (a, a, a, …) has a as its limit
* If a sequence has a limit, the limit is unique
* The sequence (1, 2, 3, …) does not have a limit

The ordinary way of proving such basic facts is via our friends Completeness and the Archimedean property. Our pal has explicitly rejected these (by affirming the existence of infinitesimals) and so does not have them available for this purpose.

To see this practically, how should we prove that the limit of the sequence (0.9, 0.99, 0.999, …) is 1? The ordinary way would be to appeal to the definition:

1. Pick any positive distance ε. By the Archimedean property, ε > 1/N > 1/10^N for some N
2. If n > N then 1-0.99…99 (with n nines) is less than 1/10^N < ε, so 1 is the limit.

The astute reader will notice this argument is very similar to the one from last week. But if infinitesimals exist, we *cannot do this*: the first step is, in fact, false. If ε is infinitesimal, then there will not be any N such that ε > 1/N! That is in fact what it means to be infinitesimal!

Specifically, if ε = 1 - 0.999…, which the Smart Man says is greater than zero, this argument falls down; we cannot get the sequence (0.9, 0.99, 0.999, …) to be ε-close to 1 if ε is infinitesimal by looking to some point far enough into the sequence: for any n, the nth term 1/10^n away from 1, and 1/10^n is larger than ε.

From further reading of SmartmanApps’ posts, I suspect he might want to object that if we continue the sequence “to infinity” the difference becomes infinitesimal. I should be very clear here: sequences as here defined and as used by him cannot be continued “to infinity”. The defining rule for this sequence is that the nth term is 1 - 1/10^n, something which makes sense and is defined for *natural numbers* n, and because infinity is not a natural number and 10^∞ is not defined, we can’t just continue like that. The only way would be to make a *definition* of what 1 - 1/10^∞ means, i.e. to *choose* what happens at this continuation; there are no axioms governing rational numbers that force us to give a certain value to this expression.

Another potential objection is that I have used the “wrong” definition of a limit, but:

1. You can find this definition in every single textbook and set of lecture notes on real analysis
2. You can find this definition (written in an old fashioned way of "variables that take on successive values" rather than sequences) in the 120-year old algebra textbooks he loves to cite
3. We saw multiple problems with his broken pseudo-definition that make it useless

so it’s up to him to provide a correct one. One could try as a first attempt to replace “for every real number greater than zero” with “for every non-infinitesimal real number greater than zero”. But without Completeness, basic facts like the uniqueness of limits, on which many more important theorems rest, would still be false.

It’s fun to explore what happens to mathematics if throw out some of its founding principles, though it does make doing anything useful with it hard. Forget working out anything truly useful like calculus without Completeness, or something precise to replace it!

Next time I plan to look a bit more at infinitesimals and how you can treat them rigorously.

#maths #math #mathematics

Should you run into this argument in your travels please feel free to share this 🧵 in article form. Every day is a good day for advancing the facts, science and data about our rapidly changing planet medium.com/@stevebentle... #ClimateChange #Biodiversity #Debunking

Debunktion Junction: “Climate ...

@fuchsi @[email protected]
"passionate about mathematical" - I'm passionate about #debunking #Gaslighters and their #bullying. He's just passionate about Gaslighting/bullying people. Note in the following how often he edited out the references which contradict him...

"The numerical value of 0.666... is a **variable** depending on the number of 6s annexed to the right". (My emphasis)" - also your EDIT, where you deliberately omitted V is the Variable being described, hence your lack of a screenshot

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Не верящий в динозавров Игорь Ашманов и отрицающий эпилепсию Василий Генералов приняты в Академию ВРАЛ

22 марта в Петербурге вручали премию за наибольший вклад в распространение лженауки в минувшем году. Звания «Почётного Академика ВРАЛ – 2025» удостоился известный IT-специалист и предприниматель, к.т.н. Игорь Ашманов.

Жюри, состоящее из ученых, присудило Игорю премию за публичные высказывания, в которых бизнесмен отрицает биологическую эволюцию, заявляет, что не верит в самопроизвольное развитие языков и «в динозавров». Победитель, по правилам премии, награждается призом — статуэткой «Грустный рептилоид». Кроме того, в финал премии вышли математик, член-корреспондент РАН Алексей Савватеев, известный выступлениями в поддержку креационизма и гомеопатии, и дизайнер Артемий Лебедев, отрицающий глобальное потепление и заявляющий, что смертность от СПИДа «это вообще ни о чем». В народном голосовании большинство голосов досталось Алексею Савватееву.

Член жюри, Академик РАН Евгений Александров выразил озабоченность тем, что лженаука проникает внутрь РАН — «престиж Академии наук важен, потому что люди доверяют ей».

«Почётным Академиком АПЧХИ» — главным распространителем лженауки в области медицины жюри из медиков признало д.м.н., невролога Василия Генералова, отрицающего эпилепсию как диагноз, продвигающего лечение аутизма у детей методами с недоказанной эффективностью и пугающего аудиторию последствиями прививок. Победитель удостоился оздоровительного приза — «Золотой кофейной клизмы». Членом-корреспондентом АПЧХИ выбрали д.м.н., хирурга Владислава Шафалинова, связывающего рост числа онкологических заболеваний с вакцинацией и электромагнитным излучением, предлагающего лечить рак «ощелачивающей терапией» и другими сомнительными методами. Врач, д.м.н. Сергей Бубновский, призывающий лечить широкий спектр заболеваний исключительно гимнастикой, не получил ни одного голоса жюри и поэтому в Академию АПЧХИ принят не был. «Приз зрительских симпатий» достался Владиславу Шафалинову.

Руководитель оргкомитета премии ВРАЛ научный журналист Александр Соколов выразил удивление по поводу того, что в финале медицинской премии впервые оказалось 3 доктора наук и задался вопросом: «Неужели ученая степень в области медицины значит так мало?» Ведущая премии, врач к.м.н. Ольга Жоголева выразила надежду, что сейчас, когда научно обоснованная медицина набирает обороты, есть шанс, что ситуация изменится к лучшему.

В рамках мероприятия состоялись выступления биолога, к.б.н. Ильи Удалова на тему «Так в чём Дарвин не прав?» и аллерголога, к.м.н. Ольги Жоголевой «Правда о БАДах».

Организатор премии — научно-просветительский портал Антропогенез.ру и проект «Ученые Против Мифов».

Среди членов жюри: академик РАН, д.ф.-м.н. Евгений Александров, д.г.н. Алексей Екайкин, д.х.н.Игорь Дмитриев, д.г.н. Ольга Соломина, д.ф.-м.н. Эмиль Ахмедов, д.г.н. Елена Сухачева, д.б.н. Тамара Кузнецова, д.и.н. Кирилл Назаренко, д.м.н. Юрий Сиволап, д.м.н. Сергей Поликарпов, к.м.н. Юлия Зинченко, к.м.н. Анна Дроганова, к.м.н. Игнат Рудченко и др.

Полный состав жюри

ВРАЛ — Вруническая Академия Лженаук, АПЧХИ — Академия Превентивной ЧакроХирургии.

Премия «Почётный Академик ВРАЛ» присуждается с 2016 года. По словам организаторов, цель премии — в шутливой форме заявить о проблеме лженауки и привлечь внимание общественности к важности борьбы с заблуждениями.

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Не верящий в динозавров Игорь Ашманов и отрицающий эпилепсию Василий Генералов приняты в Академию ВРАЛ

22 марта в Петербурге вручали премию за наибольший вклад в распространение лженауки в минувшем году. Звания «Почётного Академика ВРАЛ – 2025» удостоился известный IT-специалист и предприниматель, к.т.н. Игорь Ашманов.

Жюри, состоящее из ученых, присудило Игорю премию за публичные высказывания, в которых бизнесмен отрицает биологическую эволюцию, заявляет, что не верит в самопроизвольное развитие языков и «в динозавров». Победитель, по правилам премии, награждается призом — статуэткой «Грустный рептилоид». Кроме того, в финал премии вышли математик, член-корреспондент РАН Алексей Савватеев, известный выступлениями в поддержку креационизма и гомеопатии, и дизайнер Артемий Лебедев, отрицающий глобальное потепление и заявляющий, что смертность от СПИДа «это вообще ни о чем». В народном голосовании большинство голосов досталось Алексею Савватееву.

Член жюри, Академик РАН Евгений Александров выразил озабоченность тем, что лженаука проникает внутрь РАН — «престиж Академии наук важен, потому что люди доверяют ей».

«Почётным Академиком АПЧХИ» — главным распространителем лженауки в области медицины жюри из медиков признало д.м.н., невролога Василия Генералова, отрицающего эпилепсию как диагноз, продвигающего лечение аутизма у детей методами с недоказанной эффективностью и пугающего аудиторию последствиями прививок. Победитель удостоился оздоровительного приза — «Золотой кофейной клизмы». Членом-корреспондентом АПЧХИ выбрали д.м.н., хирурга Владислава Шафалинова, связывающего рост числа онкологических заболеваний с вакцинацией и электромагнитным излучением, предлагающего лечить рак «ощелачивающей терапией» и другими сомнительными методами. Врач, д.м.н. Сергей Бубновский, призывающий лечить широкий спектр заболеваний исключительно гимнастикой, не получил ни одного голоса жюри и поэтому в Академию АПЧХИ принят не был. «Приз зрительских симпатий» достался Владиславу Шафалинову.

Руководитель оргкомитета премии ВРАЛ научный журналист Александр Соколов выразил удивление по поводу того, что в финале медицинской премии впервые оказалось 3 доктора наук и задался вопросом: «Неужели ученая степень в области медицины значит так мало?» Ведущая премии, врач к.м.н. Ольга Жоголева выразила надежду, что сейчас, когда научно обоснованная медицина набирает обороты, есть шанс, что ситуация изменится к лучшему.

В рамках мероприятия состоялись выступления биолога, к.б.н. Ильи Удалова на тему «Так в чём Дарвин не прав?» и аллерголога, к.м.н. Ольги Жоголевой «Правда о БАДах».

Организатор премии — научно-просветительский портал Антропогенез.ру и проект «Ученые Против Мифов».

Среди членов жюри: академик РАН, д.ф.-м.н. Евгений Александров, д.г.н. Алексей Екайкин, д.х.н.Игорь Дмитриев, д.г.н. Ольга Соломина, д.ф.-м.н. Эмиль Ахмедов, д.г.н. Елена Сухачева, д.б.н. Тамара Кузнецова, д.и.н. Кирилл Назаренко, д.м.н. Юрий Сиволап, д.м.н. Сергей Поликарпов, к.м.н. Юлия Зинченко, к.м.н. Анна Дроганова, к.м.н. Игнат Рудченко и др.

Полный состав жюри

ВРАЛ — Вруническая Академия Лженаук, АПЧХИ — Академия Превентивной ЧакроХирургии.

Премия «Почётный Академик ВРАЛ» присуждается с 2016 года. По словам организаторов, цель премии — в шутливой форме заявить о проблеме лженауки и привлечь внимание общественности к важности борьбы с заблуждениями.

Пресс-релиз
Официальный сайт премии
Подробно о финалистах
Полная видеозапись мероприятия
Главные вопросы о премии ВРАЛ

Наши партнеры: SciTopus MedIQ#ВРАЛ #Антропогенез #УченыеПротивМифов #Лженаука #Псевдонаука #НаучноеМышление #Скептицизм #КритическоеМышление #НаучПросвет #ScienceCommunication #Darwin #Evolution #EvidenceBasedMedicine #AntiPseudoscience #FactChecking #ScienceEducation #RAN #Ashmanov #Generalov #ЛженаучныеМифы #ПремияВРАЛ #ScienceVsMyths #ScientificMethod #PublicScience #Medicine #Biology #Vaccination #EvidenceBased #Debunking

Не верящий в динозавров Игорь Ашманов и отрицающий эпилепсию Василий Генералов приняты в Академию ВРАЛ

22 марта в Петербурге вручали премию за наибольший вклад в распространение лженауки в минувшем году. Звания «Почётного Академика ВРАЛ – 2025» удостоился известный IT-специалист и предприниматель, к.т.н. Игорь Ашманов.

Жюри, состоящее из ученых, присудило Игорю премию за публичные высказывания, в которых бизнесмен отрицает биологическую эволюцию, заявляет, что не верит в самопроизвольное развитие языков и «в динозавров». Победитель, по правилам премии, награждается призом — статуэткой «Грустный рептилоид». Кроме того, в финал премии вышли математик, член-корреспондент РАН Алексей Савватеев, известный выступлениями в поддержку креационизма и гомеопатии, и дизайнер Артемий Лебедев, отрицающий глобальное потепление и заявляющий, что смертность от СПИДа «это вообще ни о чем». В народном голосовании большинство голосов досталось Алексею Савватееву.

Член жюри, Академик РАН Евгений Александров выразил озабоченность тем, что лженаука проникает внутрь РАН — «престиж Академии наук важен, потому что люди доверяют ей».

«Почётным Академиком АПЧХИ» — главным распространителем лженауки в области медицины жюри из медиков признало д.м.н., невролога Василия Генералова, отрицающего эпилепсию как диагноз, продвигающего лечение аутизма у детей методами с недоказанной эффективностью и пугающего аудиторию последствиями прививок. Победитель удостоился оздоровительного приза — «Золотой кофейной клизмы». Членом-корреспондентом АПЧХИ выбрали д.м.н., хирурга Владислава Шафалинова, связывающего рост числа онкологических заболеваний с вакцинацией и электромагнитным излучением, предлагающего лечить рак «ощелачивающей терапией» и другими сомнительными методами. Врач, д.м.н. Сергей Бубновский, призывающий лечить широкий спектр заболеваний исключительно гимнастикой, не получил ни одного голоса жюри и поэтому в Академию АПЧХИ принят не был. «Приз зрительских симпатий» достался Владиславу Шафалинову.

Руководитель оргкомитета премии ВРАЛ научный журналист Александр Соколов выразил удивление по поводу того, что в финале медицинской премии впервые оказалось 3 доктора наук и задался вопросом: «Неужели ученая степень в области медицины значит так мало?» Ведущая премии, врач к.м.н. Ольга Жоголева выразила надежду, что сейчас, когда научно обоснованная медицина набирает обороты, есть шанс, что ситуация изменится к лучшему.

В рамках мероприятия состоялись выступления биолога, к.б.н. Ильи Удалова на тему «Так в чём Дарвин не прав?» и аллерголога, к.м.н. Ольги Жоголевой «Правда о БАДах».

Организатор премии — научно-просветительский портал Антропогенез.ру и проект «Ученые Против Мифов».

Среди членов жюри: академик РАН, д.ф.-м.н. Евгений Александров, д.г.н. Алексей Екайкин, д.х.н.Игорь Дмитриев, д.г.н. Ольга Соломина, д.ф.-м.н. Эмиль Ахмедов, д.г.н. Елена Сухачева, д.б.н. Тамара Кузнецова, д.и.н. Кирилл Назаренко, д.м.н. Юрий Сиволап, д.м.н. Сергей Поликарпов, к.м.н. Юлия Зинченко, к.м.н. Анна Дроганова, к.м.н. Игнат Рудченко и др.

Полный состав жюри

ВРАЛ — Вруническая Академия Лженаук, АПЧХИ — Академия Превентивной ЧакроХирургии.

Премия «Почётный Академик ВРАЛ» присуждается с 2016 года. По словам организаторов, цель премии — в шутливой форме заявить о проблеме лженауки и привлечь внимание общественности к важности борьбы с заблуждениями.

Пресс-релиз
Официальный сайт премии
Подробно о финалистах
Полная видеозапись мероприятия
Главные вопросы о премии ВРАЛ

Наши партнеры: SciTopus MedIQ#ВРАЛ #Антропогенез #УченыеПротивМифов #Лженаука #Псевдонаука #НаучноеМышление #Скептицизм #КритическоеМышление #НаучПросвет #ScienceCommunication #Darwin #Evolution #EvidenceBasedMedicine #AntiPseudoscience #FactChecking #ScienceEducation #RAN #Ashmanov #Generalov #ЛженаучныеМифы #ПремияВРАЛ #ScienceVsMyths #ScientificMethod #PublicScience #Medicine #Biology #Vaccination #EvidenceBased #Debunking

Не верящий в динозавров Игорь Ашманов и отрицающий эпилепсию Василий Генералов приняты в Академию ВРАЛ

22 марта в Петербурге вручали премию за наибольший вклад в распространение лженауки в минувшем году. Звания «Почётного Академика ВРАЛ – 2025» удостоился известный IT-специалист и предприниматель, к.т.н. Игорь Ашманов.

Жюри, состоящее из ученых, присудило Игорю премию за публичные высказывания, в которых бизнесмен отрицает биологическую эволюцию, заявляет, что не верит в самопроизвольное развитие языков и «в динозавров». Победитель, по правилам премии, награждается призом — статуэткой «Грустный рептилоид». Кроме того, в финал премии вышли математик, член-корреспондент РАН Алексей Савватеев, известный выступлениями в поддержку креационизма и гомеопатии, и дизайнер Артемий Лебедев, отрицающий глобальное потепление и заявляющий, что смертность от СПИДа «это вообще ни о чем». В народном голосовании большинство голосов досталось Алексею Савватееву.

Член жюри, Академик РАН Евгений Александров выразил озабоченность тем, что лженаука проникает внутрь РАН — «престиж Академии наук важен, потому что люди доверяют ей».

«Почётным Академиком АПЧХИ» — главным распространителем лженауки в области медицины жюри из медиков признало д.м.н., невролога Василия Генералова, отрицающего эпилепсию как диагноз, продвигающего лечение аутизма у детей методами с недоказанной эффективностью и пугающего аудиторию последствиями прививок. Победитель удостоился оздоровительного приза — «Золотой кофейной клизмы». Членом-корреспондентом АПЧХИ выбрали д.м.н., хирурга Владислава Шафалинова, связывающего рост числа онкологических заболеваний с вакцинацией и электромагнитным излучением, предлагающего лечить рак «ощелачивающей терапией» и другими сомнительными методами. Врач, д.м.н. Сергей Бубновский, призывающий лечить широкий спектр заболеваний исключительно гимнастикой, не получил ни одного голоса жюри и поэтому в Академию АПЧХИ принят не был. «Приз зрительских симпатий» достался Владиславу Шафалинову.

Руководитель оргкомитета премии ВРАЛ научный журналист Александр Соколов выразил удивление по поводу того, что в финале медицинской премии впервые оказалось 3 доктора наук и задался вопросом: «Неужели ученая степень в области медицины значит так мало?» Ведущая премии, врач к.м.н. Ольга Жоголева выразила надежду, что сейчас, когда научно обоснованная медицина набирает обороты, есть шанс, что ситуация изменится к лучшему.

В рамках мероприятия состоялись выступления биолога, к.б.н. Ильи Удалова на тему «Так в чём Дарвин не прав?» и аллерголога, к.м.н. Ольги Жоголевой «Правда о БАДах».

Организатор премии — научно-просветительский портал Антропогенез.ру и проект «Ученые Против Мифов».

Среди членов жюри: академик РАН, д.ф.-м.н. Евгений Александров, д.г.н. Алексей Екайкин, д.х.н.Игорь Дмитриев, д.г.н. Ольга Соломина, д.ф.-м.н. Эмиль Ахмедов, д.г.н. Елена Сухачева, д.б.н. Тамара Кузнецова, д.и.н. Кирилл Назаренко, д.м.н. Юрий Сиволап, д.м.н. Сергей Поликарпов, к.м.н. Юлия Зинченко, к.м.н. Анна Дроганова, к.м.н. Игнат Рудченко и др.

Полный состав жюри

ВРАЛ — Вруническая Академия Лженаук, АПЧХИ — Академия Превентивной ЧакроХирургии.

Премия «Почётный Академик ВРАЛ» присуждается с 2016 года. По словам организаторов, цель премии — в шутливой форме заявить о проблеме лженауки и привлечь внимание общественности к важности борьбы с заблуждениями.

Пресс-релиз
Официальный сайт премии
Подробно о финалистах
Полная видеозапись мероприятия
Главные вопросы о премии ВРАЛ

Наши партнеры: SciTopus MedIQ#ВРАЛ #Антропогенез #УченыеПротивМифов #Лженаука #Псевдонаука #НаучноеМышление #Скептицизм #КритическоеМышление #НаучПросвет #ScienceCommunication #Darwin #Evolution #EvidenceBasedMedicine #AntiPseudoscience #FactChecking #ScienceEducation #RAN #Ashmanov #Generalov #ЛженаучныеМифы #ПремияВРАЛ #ScienceVsMyths #ScientificMethod #PublicScience #Medicine #Biology #Vaccination #EvidenceBased #Debunking

Не верящий в динозавров Игорь Ашманов и отрицающий эпилепсию Василий Генералов приняты в Академию ВРАЛ

22 марта в Петербурге вручали премию за наибольший вклад в распространение лженауки в минувшем году. Звания «Почётного Академика ВРАЛ – 2025» удостоился известный IT-специалист и предприниматель, к.т.н. Игорь Ашманов.

Жюри, состоящее из ученых, присудило Игорю премию за публичные высказывания, в которых бизнесмен отрицает биологическую эволюцию, заявляет, что не верит в самопроизвольное развитие языков и «в динозавров». Победитель, по правилам премии, награждается призом — статуэткой «Грустный рептилоид». Кроме того, в финал премии вышли математик, член-корреспондент РАН Алексей Савватеев, известный выступлениями в поддержку креационизма и гомеопатии, и дизайнер Артемий Лебедев, отрицающий глобальное потепление и заявляющий, что смертность от СПИДа «это вообще ни о чем». В народном голосовании большинство голосов досталось Алексею Савватееву.

Член жюри, Академик РАН Евгений Александров выразил озабоченность тем, что лженаука проникает внутрь РАН — «престиж Академии наук важен, потому что люди доверяют ей».

«Почётным Академиком АПЧХИ» — главным распространителем лженауки в области медицины жюри из медиков признало д.м.н., невролога Василия Генералова, отрицающего эпилепсию как диагноз, продвигающего лечение аутизма у детей методами с недоказанной эффективностью и пугающего аудиторию последствиями прививок. Победитель удостоился оздоровительного приза — «Золотой кофейной клизмы». Членом-корреспондентом АПЧХИ выбрали д.м.н., хирурга Владислава Шафалинова, связывающего рост числа онкологических заболеваний с вакцинацией и электромагнитным излучением, предлагающего лечить рак «ощелачивающей терапией» и другими сомнительными методами. Врач, д.м.н. Сергей Бубновский, призывающий лечить широкий спектр заболеваний исключительно гимнастикой, не получил ни одного голоса жюри и поэтому в Академию АПЧХИ принят не был. «Приз зрительских симпатий» достался Владиславу Шафалинову.

Руководитель оргкомитета премии ВРАЛ научный журналист Александр Соколов выразил удивление по поводу того, что в финале медицинской премии впервые оказалось 3 доктора наук и задался вопросом: «Неужели ученая степень в области медицины значит так мало?» Ведущая премии, врач к.м.н. Ольга Жоголева выразила надежду, что сейчас, когда научно обоснованная медицина набирает обороты, есть шанс, что ситуация изменится к лучшему.

В рамках мероприятия состоялись выступления биолога, к.б.н. Ильи Удалова на тему «Так в чём Дарвин не прав?» и аллерголога, к.м.н. Ольги Жоголевой «Правда о БАДах».

Организатор премии — научно-просветительский портал Антропогенез.ру и проект «Ученые Против Мифов».

Среди членов жюри: академик РАН, д.ф.-м.н. Евгений Александров, д.г.н. Алексей Екайкин, д.х.н.Игорь Дмитриев, д.г.н. Ольга Соломина, д.ф.-м.н. Эмиль Ахмедов, д.г.н. Елена Сухачева, д.б.н. Тамара Кузнецова, д.и.н. Кирилл Назаренко, д.м.н. Юрий Сиволап, д.м.н. Сергей Поликарпов, к.м.н. Юлия Зинченко, к.м.н. Анна Дроганова, к.м.н. Игнат Рудченко и др.

Полный состав жюри

ВРАЛ — Вруническая Академия Лженаук, АПЧХИ — Академия Превентивной ЧакроХирургии.

Премия «Почётный Академик ВРАЛ» присуждается с 2016 года. По словам организаторов, цель премии — в шутливой форме заявить о проблеме лженауки и привлечь внимание общественности к важности борьбы с заблуждениями.

Пресс-релиз
Официальный сайт премии
Подробно о финалистах
Полная видеозапись мероприятия
Главные вопросы о премии ВРАЛ

Наши партнеры: SciTopus MedIQ#ВРАЛ #Антропогенез #УченыеПротивМифов #Лженаука #Псевдонаука #НаучноеМышление #Скептицизм #КритическоеМышление #НаучПросвет #ScienceCommunication #Darwin #Evolution #EvidenceBasedMedicine #AntiPseudoscience #FactChecking #ScienceEducation #RAN #Ashmanov #Generalov #ЛженаучныеМифы #ПремияВРАЛ #ScienceVsMyths #ScientificMethod #PublicScience #Medicine #Biology #Vaccination #EvidenceBased #Debunking

Décryptage essentiel! Extrait du reportage "PIECES À CONVICTION" qui explore l'usage du Rivotril en Ehpad et démonte les idées reçues. Court, percutant et utile pour mieux saisir les enjeux médicaux et éthiques. À regarder et partager pour nourrir le débat. #Debunking #Rivotril #EHPAD #COVID19 #Reportage #FactCheck #Science #French
https://video.voiceover.bar/videos/watch/227e173c-69a4-44fc-8494-53f197694263

Décryptage essentiel! Extrait du reportage "PIECES À CONVICTION" qui explore l'usage du Rivotril en Ehpad et démonte les idées reçues. Court, percutant et utile pour mieux saisir les enjeux médicaux et éthiques. À regarder et partager pour nourrir le débat. #Debunking #Rivotril #EHPAD #COVID19 #Reportage #FactCheck #Science #French
https://video.voiceover.bar/videos/watch/227e173c-69a4-44fc-8494-53f197694263

Debunking Zswap and Zram Myths

https://chrisdown.name/2026/03/24/zswap-vs-zram-when-to-use-what.html

#HackerNews #Debunking #Zswap #and #Zram #Myths #Zswap #Zram #Myths #Linux #Performance #Optimization

This AI Image Detector Checks a Global Database of Debunked Pictures https://petapixel.com/2026/02/23/ai-image-whisperer-detector-debunked/ #reverseimagesearch #aiimagedetection #imagedetector #Technology #provenance #debunking #fakephoto #aiimage #verify #genai #News

This AI Image Detector Checks a Global Database of Debunked Pictures https://petapixel.com/2026/02/23/ai-image-whisperer-detector-debunked/ #reverseimagesearch #aiimagedetection #imagedetector #Technology #provenance #debunking #fakephoto #aiimage #verify #genai #News

This AI Image Detector Checks a Global Database of Debunked Pictures https://petapixel.com/2026/02/23/ai-image-whisperer-detector-debunked/ #reverseimagesearch #aiimagedetection #imagedetector #Technology #provenance #debunking #fakephoto #aiimage #verify #genai #News

This AI Image Detector Checks a Global Database of Debunked Pictures https://petapixel.com/2026/02/23/ai-image-whisperer-detector-debunked/ #reverseimagesearch #aiimagedetection #imagedetector #Technology #provenance #debunking #fakephoto #aiimage #verify #genai #News

This AI Image Detector Checks a Global Database of Debunked Pictures https://petapixel.com/2026/02/23/ai-image-whisperer-detector-debunked/ #reverseimagesearch #aiimagedetection #imagedetector #Technology #provenance #debunking #fakephoto #aiimage #verify #genai #News

A #LineaRadioSavona, oggi, con GianFunk abbiamo investigato sul #Luanniao, portaerei #spaziale tipo "Blue Noah" che gira dal 2019 nei progetti cinesi!
Grazie a chi ha ascoltato, buon sabato!🙏👋🚀
#UnoRadio #Spazio
#MastoRadio #Debunking @astronomia

A #LineaRadioSavona, oggi, con GianFunk abbiamo investigato sul #Luanniao, portaerei #spaziale tipo "Blue Noah" che gira dal 2019 nei progetti cinesi!
Grazie a chi ha ascoltato, buon sabato!🙏👋🚀
#UnoRadio #Spazio
#MastoRadio #Debunking @astronomia

#RickDufer insulta #Mortebianca mentre parla di #Trump ed #Epstein qui il #Debunking completo

https://www.youtube.com/watch?v=e_4mA45o-XU

#RickDufer insulta #Mortebianca mentre parla di #Trump ed #Epstein qui il #Debunking completo

https://www.youtube.com/watch?v=e_4mA45o-XU

Since this is starting to look like a bad week for US fascism, I might as well pile on with a #scicomm 🧵 .

What I'll attempt here is a #debunking of last year's [briefly] notorious Ethics & Public Policy Center (EPPC) purported analysis of mifepristone adverse events in a healthcare claims database. 1/n
https://eppc.org/stop-harming-women/

Im Blick auf das debanking interessant: >>[...] Auch dieses Argument hören Sparbuchinhaber oft. Der Bundesgerichtshof (BGH, Urt. v. 4.6.2002; Az: XI ZR 361/01) hat in diesem Zusammenhang entschieden, dass eine Verwirkung des Auszahlungsanspruches gegen die Bank wegen jahrzehntelanger Untätigkeit nicht eintreten kann. Die Bank kann nicht darauf vertrauen, nicht mehr in Anspruch genommen zu werden. Ebenso hat der BGH in diesem Urteil entschieden, dass bei einem beidseitig kündbaren Sparkonto die Verjährungsfrist erst dann zu laufen beginnt, wenn von einer der Vertragsparteien eine Kündigung ausgesprochen wurde. [...]<< https://www.vis.bayern.de/geld_versicherungen/finanzanlagen_altersvorsorge/sparbuch_vergessen.htm
#debunking #gls #sparkasse #sparbuch

Saturday, December 20, 2025

https://activitypub.writeworks.uk/2025/12/saturday-december-20-2025/

Saturday, December 20, 2025

https://activitypub.writeworks.uk/2025/12/saturday-december-20-2025/

Saturday, December 20, 2025

https://activitypub.writeworks.uk/2025/12/saturday-december-20-2025/

Saturday, December 20, 2025

https://activitypub.writeworks.uk/2025/12/saturday-december-20-2025/

Saturday, December 20, 2025

https://activitypub.writeworks.uk/2025/12/saturday-december-20-2025/

Consumers share the common 'money-saving' habits that actually cost them more over time

https://fed.brid.gy/r/https://www.upworthy.com/false-money-saving-tactics

#Angola: #Debunking the #Myths of the #Lobito #Corridor https://allafrica.com/stories/202512030481.html

In truth, it is a mirror of everything negative the continent endures: Chinese debt, Western opportunism, Congolese blood, Angolan misrule...

#Angola: #Debunking the #Myths of the #Lobito #Corridor https://allafrica.com/stories/202512030481.html

In truth, it is a mirror of everything negative the continent endures: Chinese debt, Western opportunism, Congolese blood, Angolan misrule...

#Angola: #Debunking the #Myths of the #Lobito #Corridor https://allafrica.com/stories/202512030481.html

In truth, it is a mirror of everything negative the continent endures: Chinese debt, Western opportunism, Congolese blood, Angolan misrule...

#Angola: #Debunking the #Myths of the #Lobito #Corridor https://allafrica.com/stories/202512030481.html

In truth, it is a mirror of everything negative the continent endures: Chinese debt, Western opportunism, Congolese blood, Angolan misrule...

#Angola: #Debunking the #Myths of the #Lobito #Corridor https://allafrica.com/stories/202512030481.html

In truth, it is a mirror of everything negative the continent endures: Chinese debt, Western opportunism, Congolese blood, Angolan misrule...

https://www.evshift.com/363851/ev-myth-busters-debunking-tesla-range-charging-myths-shorts/ EV Myth Busters: Debunking Tesla Range & Charging Myths #shorts #BUSTERS #charging #Debunking #ElectricCars #ElectricVehicles #EV #Myth #Myths #Range #shorts #Tesla

The #Bible Doesn't Say So

https://www.youtube.com/watch?v=VidaQPd3C6Q

Dr. Dan McClellan is well known for #debunking #Biblical #misinformation online. He’ll explain why understanding biases matters and how misunderstanding #ancienttexts shapes modern debates in harmful ways. Dr. Dan McClellan is a public scholar of the Bible and #religion. He earned his PhD in #theology and religion from the University of Exeter

Une présentation du livre "Nos ancêtres les pharaons. Cinq siècles d'illusions sur l'Égypte ancienne", par Jean-Loïc Le Quellec, anthropologue et directeur de recherche au CNRS. Une plongée vertigineuse dans les délires complotistes et pseudoscientifiques autour de l'Égypte ancienne, qu'il débunke avec une patience salutaire.
Sur le blog de lecture Hugin & Munin (blog d'inspirations rôlistiques).
https://hu-mu.blogspot.com/2025/10/nos-ancetres-les-pharaons-par-jean-loic.html
#Antiquite #EgypteAncienne #pseudoscience #complotisme #ésotérisme #AfriqueAncienne #Australie #Mayas #CivilisationsPrécolombiennes #Gims #Youtube #anthropologie #vulgarisation #debunking

Une présentation du livre "Nos ancêtres les pharaons. Cinq siècles d'illusions sur l'Égypte ancienne", par Jean-Loïc Le Quellec, anthropologue et directeur de recherche au CNRS. Une plongée vertigineuse dans les délires complotistes et pseudoscientifiques autour de l'Égypte ancienne, qu'il débunke avec une patience salutaire.
Sur le blog de lecture Hugin & Munin (blog d'inspirations rôlistiques).
https://hu-mu.blogspot.com/2025/10/nos-ancetres-les-pharaons-par-jean-loic.html
#Antiquite #EgypteAncienne #pseudoscience #complotisme #ésotérisme #AfriqueAncienne #Australie #Mayas #CivilisationsPrécolombiennes #Gims #Youtube #anthropologie #vulgarisation #debunking

Une présentation du livre "Nos ancêtres les pharaons. Cinq siècles d'illusions sur l'Égypte ancienne", par Jean-Loïc Le Quellec, anthropologue et directeur de recherche au CNRS. Une plongée vertigineuse dans les délires complotistes et pseudoscientifiques autour de l'Égypte ancienne, qu'il débunke avec une patience salutaire.
Sur le blog de lecture Hugin & Munin (blog d'inspirations rôlistiques).
https://hu-mu.blogspot.com/2025/10/nos-ancetres-les-pharaons-par-jean-loic.html
#Antiquite #EgypteAncienne #pseudoscience #complotisme #ésotérisme #AfriqueAncienne #Australie #Mayas #CivilisationsPrécolombiennes #Gims #Youtube #anthropologie #vulgarisation #debunking