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FQXi FORUM

October 18, 2019

CATEGORY:
Trick or Truth Essay Contest (2015)
[back]

TOPIC: Mathematics of Science by Robert MacDuff [refresh]

TOPIC: Mathematics of Science by Robert MacDuff [refresh]

The author addresses the question: Is it possible to integrate math and science? The metaphor of divorce is used to distinguish mathematics’ reliance on quantitative methods from science’s requirements for tools to encode structural and quantitative relationships. The reader’s attention is engaged by inviting them to experience varying degrees of cognitive dissonance as they struggle to understand quantitatively, problems that require the integration of both. The reader is guided, through the introduction of structurally based axioms, into the development of a mathematics where the symbols have quantitative and structural referents.

Robert C. (Rob) MacDuff, Ph.D., holds degrees in physics and mathematics education. His interests are in physics, mathematics, philosophy and neuroscience, more specifically the development of conceptual mental tools and their application to the learning of math: Cognitive Instruction in Mathematics Modeling (CIMM) and physics: Cognitive Instruction in Modeling Physics (CIMP).

Millennia ago math and physics were part of philosophy. With increased volume of knowledge, the people specialized and grew apart. Each specialty assumed a world (a guild) of their own. These guilds have some natural components but increasingly include many invented characteristics. For example, physics has been traditionally a cause- effect world. But math has invented statistics and then QM, which is unreal. Now we discuss the invented/discovered characteristics of both. Synthesis is difficult.

The separation is not so much a divorce as a parting of the ways of siblings.

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The separation is not so much a divorce as a parting of the ways of siblings.

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I agree “the parting of the ways” is philosophical as the difference shows up in the definitions and axioms. However, which metaphor one chooses to describe the parting of the ways, parents or siblings, is not so important as understanding its consequences. The lack of a sense of reality you mention is due to mathematics dropping units, which removed intuition, laws, structure, etc. Elimination of units also dropped metaphorical, analogical and transitive reasoning processes for rule-based axiomatic or algebraic reasoning.

As you have pointed out, mathematics has created a stunning array of mathematical systems based upon relationally connected sets and yet, it can’t model something so simple as an array of four by three dots. It is capable of computing an answer to which a dot can be appended. It is the structure of the array that can’t be modeled by a serial order (quantitative) based systems. Three times four may compute to twelve, but three what and four what will produce 12 dots.

As you have pointed out, mathematics has created a stunning array of mathematical systems based upon relationally connected sets and yet, it can’t model something so simple as an array of four by three dots. It is capable of computing an answer to which a dot can be appended. It is the structure of the array that can’t be modeled by a serial order (quantitative) based systems. Three times four may compute to twelve, but three what and four what will produce 12 dots.

Rob

This paper is stunning, a breakthrough in understanding the reasoning processes necessary for science. Even though I helped edit your paper, I still struggle with the simplicity of it. Here is why I think your paper is destined to be seen as a seminal work. It introduces a whole series of major ideas, including:

1) The introduction of a law of the included middle into...

view entire post

This paper is stunning, a breakthrough in understanding the reasoning processes necessary for science. Even though I helped edit your paper, I still struggle with the simplicity of it. Here is why I think your paper is destined to be seen as a seminal work. It introduces a whole series of major ideas, including:

1) The introduction of a law of the included middle into...

view entire post

report post as inappropriate

Thank you Chris for your assistance, kind words and encouragement. You mentioned the simplicity of it. Yes, it seems simple but yet it does all standard physics and I believe it can be extended to special and general relativity. In addition, it works wonderfully in the solution of logic problems. So yes it is simple, but not too simple.

Joe I hope you don’t mind if I both agree and disagree with you. Your statement: “… I think that abstract mathematics and abstract physics have nothing to do with the real Universe …” is in some sense correct. However I disagree with your reasons and I think that if you try this exercise you too will disagree. Take a bunch (8) of checkers © and put them into groups, rearrange them into different groups and do this again and again until you see a pattern. Suddenly it will occur to you that the different groups are the same but not the same. They are grouped differently but the total number never changes. E.g. 1/ 5© + 3© = 8©, 2/ 2© + (1© + 5©) = (2© + 1©) + 5© or 3/ 2(3© +1©) = 2(3©) + 2(1©). What you have discovered is the law of invariance under regrouping and the mathematical rules for encoding it and these rules only apply to objects that behave in this way.

Now I think you will agree that we can use symbols to describe exactly what we observe and that is precisely what this paper is all about. However, the application of the law of the included middle is required in order to perceive the patterns.

Enjoy

Rob

Now I think you will agree that we can use symbols to describe exactly what we observe and that is precisely what this paper is all about. However, the application of the law of the included middle is required in order to perceive the patterns.

Enjoy

Rob

Dear Dr. MacDuff,

I mean no disrespect, but reality is not optional. One can be abstractly correct or abstractly incorrect only about an abstraction.

Indubitably, I have a complete real skin surface. Every person, place and thing has a complete real surface of one sort of another. Each real checker has a complete real surface. Instead of idly trying to construct abstract patterns from an abstract group of abstract checkers, simply understand that the surface of every real checker travels at the same constant speed as the surface of everything else that is real.

Thank you for not reporting my comment as being inappropriate and having it declared Obnoxious Spam by FQXi.org

Joe Fisher

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I mean no disrespect, but reality is not optional. One can be abstractly correct or abstractly incorrect only about an abstraction.

Indubitably, I have a complete real skin surface. Every person, place and thing has a complete real surface of one sort of another. Each real checker has a complete real surface. Instead of idly trying to construct abstract patterns from an abstract group of abstract checkers, simply understand that the surface of every real checker travels at the same constant speed as the surface of everything else that is real.

Thank you for not reporting my comment as being inappropriate and having it declared Obnoxious Spam by FQXi.org

Joe Fisher

report post as inappropriate

I have some questions! I'm a high school Physics teacher and I recognize in this paper the difference between how my students experience numbers and how I want them to... realizing that for them, their numerical "answer" is an entity unto itself kind of makes it a little bit more understandable how they constantly leave off units or fail to interpret their calculations!

So overall I think I...

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So overall I think I...

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Lucas, you are perfectly correct in assuming that many of these ideas will ease instruction. Your comments reflect, to some degree, my own struggles to understand.

1) Numbers are used in three different ways: ordinal/relational, cardinal/quantitative and relationship/structural. In mathematics the latter concept is tacitly used to construct numerical symbols but from then on they are...

view entire post

1) Numbers are used in three different ways: ordinal/relational, cardinal/quantitative and relationship/structural. In mathematics the latter concept is tacitly used to construct numerical symbols but from then on they are...

view entire post

Rob,

I think many if not most mathematicians consider pedagogical reform one of the most critical issues in mathematics today. Maybe we don't need to start with noumena and phenomena, but we certainly do need to see the fun in mathematics before it's beaten out of us by fourth grade.

And I think that there may be something inherent in our natures, as Kant thought, that 'gets' it -- without being beaten and stressed. As my then six-year old granddaughter told her mother, "If all numbers go on and on forever without stopping, then all numbers are small numbers." Leibniz would have been proud, and so would Hermann Weyl -- your reference is my second favorite Weyl product (the first is *The Continuum.*).

Hope you get a chance to visit my essay. Yours gets my highest mark and best wishes for success.

Best,

Tom

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I think many if not most mathematicians consider pedagogical reform one of the most critical issues in mathematics today. Maybe we don't need to start with noumena and phenomena, but we certainly do need to see the fun in mathematics before it's beaten out of us by fourth grade.

And I think that there may be something inherent in our natures, as Kant thought, that 'gets' it -- without being beaten and stressed. As my then six-year old granddaughter told her mother, "If all numbers go on and on forever without stopping, then all numbers are small numbers." Leibniz would have been proud, and so would Hermann Weyl -- your reference is my second favorite Weyl product (the first is *The Continuum.*).

Hope you get a chance to visit my essay. Yours gets my highest mark and best wishes for success.

Best,

Tom

report post as inappropriate

Tom,

I totally agree on your astute awareness of the critical issue of pedagogical reform. Already a third of students have been lost by grade 3 and 85% by high school. The destruction of human potential is extreme. The difficulty is that we don’t teach math we teach computation. We teach axioms not model construction. We teach operations not concept construction. The question as to what is a number is rarely if ever addressed.

I have low hopes for an integrated math/science program anytime soon.

By the way, I love your reference to mathematics as being a religion.

thanks

Rob

I totally agree on your astute awareness of the critical issue of pedagogical reform. Already a third of students have been lost by grade 3 and 85% by high school. The destruction of human potential is extreme. The difficulty is that we don’t teach math we teach computation. We teach axioms not model construction. We teach operations not concept construction. The question as to what is a number is rarely if ever addressed.

I have low hopes for an integrated math/science program anytime soon.

By the way, I love your reference to mathematics as being a religion.

thanks

Rob

Hi Rob,

Your reference to your personal blog ("What is one quarter plus one quarter?") got me thinking -- how often have those feigning intellectual superiority compared another's "crackpot thinking" to a quest for the impossible construction "squaring the circle?"

It's as if they deem empirically-based technical activities -- such as applying the compass and straightedge, and...

view entire post

Your reference to your personal blog ("What is one quarter plus one quarter?") got me thinking -- how often have those feigning intellectual superiority compared another's "crackpot thinking" to a quest for the impossible construction "squaring the circle?"

It's as if they deem empirically-based technical activities -- such as applying the compass and straightedge, and...

view entire post

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Tom, I am wondering if you have missed the point of the circle/square problem. May I be so bold to say that there is this “religious” (your term) belief in the power of mathematics and yet it is unable to solve such an incredibly simple problem. Once noticed, all of a sudden there is this infinite variety of simple structures current mathematical programs are unable to model.

Complex numbers with real and imaginary axis may seem delightful toys. The true delight occurs when you can perceive and reason about the real world and the word imaginary slips off into oblivion.

Let me put it another way, I think that when mathematicians chose the path of rigor over intuitionism (Hilbert vs Brouwer) they threw the baby out with the bath water. Boole did the same to logic. Seriously Tom, would you not like to be able to do logic in exactly the same way that you do algebra? Write out the equations and algebraically deduce the answer. No, I guess not, as that would not be “merely recreational.”

There are many similarities between David and myself; however, there are also significant differences.

As to Kant, far beyond brilliant, I am always stunned by what he was able to do. Hume as well, and to call him an empiricist is downright cruel. I believe that he is the that lead Kant to the concept of synthetic a priori.

And Tom your posts are a delight

cheers

Rob

Complex numbers with real and imaginary axis may seem delightful toys. The true delight occurs when you can perceive and reason about the real world and the word imaginary slips off into oblivion.

Let me put it another way, I think that when mathematicians chose the path of rigor over intuitionism (Hilbert vs Brouwer) they threw the baby out with the bath water. Boole did the same to logic. Seriously Tom, would you not like to be able to do logic in exactly the same way that you do algebra? Write out the equations and algebraically deduce the answer. No, I guess not, as that would not be “merely recreational.”

There are many similarities between David and myself; however, there are also significant differences.

As to Kant, far beyond brilliant, I am always stunned by what he was able to do. Hume as well, and to call him an empiricist is downright cruel. I believe that he is the that lead Kant to the concept of synthetic a priori.

And Tom your posts are a delight

cheers

Rob

Rob, Painleve' is credited with saying, "The shortest path between two truths in the real domain passes through the complex domain."

I'm very much an admirer of Brouwer (and Weierstrass, Dedekind, Weyl) -- ("all real functions of a real valued variable are continuous") -- as well as Hilbert. I like the Formalists and I like the Intuitionists equally, I think, for different reasons.

Hilbert makes me feel secure about specifying boundary conditions for complex functions in a space I can understand, if not physically experience. Brouwer's "twoity" of functions, and every mathematical act as a "move of time", give me permission to superpose myself, so to speak, within the world of objects as an element in relation. Equality, transitivity, reflexivity become palpable -- so would I like to do logic the way I do algebra? I think I already do. Formal logic, after all, is only a branch of mathematical theory; it is not a theorem in the mathematical canon, such as the fundamental theorem of algebra. I am thinking of that closing line by Doc Brown to Marty in *Back to the Future*: "Roads? Where we're going we don't need roads."

Logic only gets one as far as the dirt path. After that come the graders and pavers, and maybe later even time machine makers. At the beginning of proving a theorem, though, one must take risks and make leaps. Barry Mazur (*Imagining Numbers*) said that when he is introduced to a new result, his initial reaction is often, "I didn't know you could do that!"

So why couldn't any 8-year old be as free as a research mathematician to take risks, make leaps, have fun, be wrong? One may then see mathematics as not so different from "real life." Work and recreation united. I agree with you about the loss of human potential.

Well, here you compliment me for posts, and then I go rambling on. I delight in your posts, too.

Best,

Tom

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I'm very much an admirer of Brouwer (and Weierstrass, Dedekind, Weyl) -- ("all real functions of a real valued variable are continuous") -- as well as Hilbert. I like the Formalists and I like the Intuitionists equally, I think, for different reasons.

Hilbert makes me feel secure about specifying boundary conditions for complex functions in a space I can understand, if not physically experience. Brouwer's "twoity" of functions, and every mathematical act as a "move of time", give me permission to superpose myself, so to speak, within the world of objects as an element in relation. Equality, transitivity, reflexivity become palpable -- so would I like to do logic the way I do algebra? I think I already do. Formal logic, after all, is only a branch of mathematical theory; it is not a theorem in the mathematical canon, such as the fundamental theorem of algebra. I am thinking of that closing line by Doc Brown to Marty in *Back to the Future*: "Roads? Where we're going we don't need roads."

Logic only gets one as far as the dirt path. After that come the graders and pavers, and maybe later even time machine makers. At the beginning of proving a theorem, though, one must take risks and make leaps. Barry Mazur (*Imagining Numbers*) said that when he is introduced to a new result, his initial reaction is often, "I didn't know you could do that!"

So why couldn't any 8-year old be as free as a research mathematician to take risks, make leaps, have fun, be wrong? One may then see mathematics as not so different from "real life." Work and recreation united. I agree with you about the loss of human potential.

Well, here you compliment me for posts, and then I go rambling on. I delight in your posts, too.

Best,

Tom

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Any reader who has appreciated Rob MacDuff's essay might want to also read "Modeling the Physical World with Common Sense and Mathematics" by David Hestenes, elsewhere on this board.

My new post to Hestenes' essay contains some further reflections on Rob's work and discusses a deeper connection that I see between the work of the two authors.

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My new post to Hestenes' essay contains some further reflections on Rob's work and discusses a deeper connection that I see between the work of the two authors.

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Dear Rob,

It seems to me that the issues you cover in your essay could well fill a whole book! I do have one question, though, and a definitive answer to it would go a long way towards illuminating questions about the merits of ideas like the mathematical universe hypothesis (MUH).

I am not a proponent of MUH, but when I tried to formulate a counterargument based on the fact that...

view entire post

It seems to me that the issues you cover in your essay could well fill a whole book! I do have one question, though, and a definitive answer to it would go a long way towards illuminating questions about the merits of ideas like the mathematical universe hypothesis (MUH).

I am not a proponent of MUH, but when I tried to formulate a counterargument based on the fact that...

view entire post

report post as inappropriate

Armin you pose a very interesting question, however I believe the answer is much simpler than you would expect. First of all we are limited to only being able to describe “this” in terms of “that”. And this is only possible if there is something defining a connection between the two.

In the case of like (the defining feature determined by us) quantities the connection between them is symbolized by a numeral. In other words numbers are ratios! What is well known in physics (not necessarily mathematics) is that ratios do not add. So I would like to ask what do you mean by 1 + 1 = 2? Suppose in your example the 1 refers to a drop of water. Then I am curious as to what you believe 1 x 1 might be. It is the answer to this latter question, which I believe is the answer to your original question.

I will have to read Sophia Magnusdottir's essay as it sounds interesting.

Cheers

Rob

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In the case of like (the defining feature determined by us) quantities the connection between them is symbolized by a numeral. In other words numbers are ratios! What is well known in physics (not necessarily mathematics) is that ratios do not add. So I would like to ask what do you mean by 1 + 1 = 2? Suppose in your example the 1 refers to a drop of water. Then I am curious as to what you believe 1 x 1 might be. It is the answer to this latter question, which I believe is the answer to your original question.

I will have to read Sophia Magnusdottir's essay as it sounds interesting.

Cheers

Rob

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Dear Rob,

Without thinking about it too deeply I would take the "meaning" of 1x1 to be any member of a set of propositions which can only be narrowed down by giving further information, because the further information determines the context in which I am using it.

If I understand correctly, one of your arguments can be paraphrased by saying that in mathematics this additional information is usually ignored by replacing the referent with the relationship it has in the ordering to other numbers, effectively representing just itself, whereas in physics it is inevitable that numbers do not just represent themselves but things and relationships between things in the world. Furthermore, this can get confusing if one does not recognize the context in which a number is used (such as pi in your example). I tend to agree with all this.

What is not so clear to me yet is how this answers the question I originally asked. In particular, one could imagine a very long list of propositions in which each describes, say, "m" in terms something else, say, "E", and include in the list all of the proposition that permit one to circle back from "E" to "m", and do this for every single "that" in terms of which "m" is described, and furthermore carefully distinguish whether in each instance a number is used as a ratio or in a purely mathematical sense. That would make the list a giant tautology, but does this really undermine the MUH? If yes, how?

Thanks,

Armin

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Without thinking about it too deeply I would take the "meaning" of 1x1 to be any member of a set of propositions which can only be narrowed down by giving further information, because the further information determines the context in which I am using it.

If I understand correctly, one of your arguments can be paraphrased by saying that in mathematics this additional information is usually ignored by replacing the referent with the relationship it has in the ordering to other numbers, effectively representing just itself, whereas in physics it is inevitable that numbers do not just represent themselves but things and relationships between things in the world. Furthermore, this can get confusing if one does not recognize the context in which a number is used (such as pi in your example). I tend to agree with all this.

What is not so clear to me yet is how this answers the question I originally asked. In particular, one could imagine a very long list of propositions in which each describes, say, "m" in terms something else, say, "E", and include in the list all of the proposition that permit one to circle back from "E" to "m", and do this for every single "that" in terms of which "m" is described, and furthermore carefully distinguish whether in each instance a number is used as a ratio or in a purely mathematical sense. That would make the list a giant tautology, but does this really undermine the MUH? If yes, how?

Thanks,

Armin

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Armin you have in fact answered your own question by realizing that the 1x1 does not have any meaning in the context that I proposed. What you will quickly discover that no matter what proposition you construct for 1 + 1 it will not work for 1x1 or 1/1 or 1-1. These operations represent four different types of structures required for developing a set of symbols to represent structure contained in the world.

MUH if I understand it correctly is assumes the existence of mathematical structure. However, as Hilbert and others realized, numbers have only one structure and that is a serial order. What is amazing is the vast interpretations of number that arise out of all the possible contexts in which it can be utilized but that doesn’t even begin to qualify for all the structures required to construct a world. My paper provides one example, which provides a template for an infinite variety of situations mathematics is incapable of modeling.

Regarding your last example, yes you may imagine a very long list of propositions. However, what you are missing is that each entry in the sequence of “this” and “that” has to be defined by something. If you decided to construct such a loop then you would be constructing a self-referential loop, upon which you might find that you are your own grandfather.

Cheers

Rob

MUH if I understand it correctly is assumes the existence of mathematical structure. However, as Hilbert and others realized, numbers have only one structure and that is a serial order. What is amazing is the vast interpretations of number that arise out of all the possible contexts in which it can be utilized but that doesn’t even begin to qualify for all the structures required to construct a world. My paper provides one example, which provides a template for an infinite variety of situations mathematics is incapable of modeling.

Regarding your last example, yes you may imagine a very long list of propositions. However, what you are missing is that each entry in the sequence of “this” and “that” has to be defined by something. If you decided to construct such a loop then you would be constructing a self-referential loop, upon which you might find that you are your own grandfather.

Cheers

Rob

Dear Rob

Thank you for writing such a thought-provoking essay.

I had the pleasure of having dinner with David Hestenes last week and we discussed your essay (along with ours) at some length.

You have convinced me that multiplication as repeated addition is not the same as multiplication of scale or ratios. While different arguments will have different effects on people, your statement that in the problem

2 + 2 + 2 = 3 x 2

the 3 on the RHS acts as an adjective and the 2 on the RHS acts as a noun, really struck me. I get it! Very subtle! Bravo!

Now, you have read my essay, so this may make some sense to you:

I *know* that you can show that the x operator above is associative and commutative which leads to it being an invertible transform of additivity. However, you can also show that x distributes over + in repeated addition, which constrains the quantification to being a log so that x must be multiplication.

Now what I bet you can show, is that the symmetries of ratios also lead to multiplication. So I am willing to bet that both problems "repeated addition" and "scaling" are quantified by the same function but for different reasons.

I am going to look into this as that would be really cool!

I also wanted to say that your circle-square problem is mesmerizing and I think that it effectively highlights the fact that there are some unrecognized subtleties still lingering in the metaphors that lead to the mathematics that we use (see Hestenes' essay and mine).

Thank you for a very enjoyable and thought-provoking essay!

Kevin Knuth

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Thank you for writing such a thought-provoking essay.

I had the pleasure of having dinner with David Hestenes last week and we discussed your essay (along with ours) at some length.

You have convinced me that multiplication as repeated addition is not the same as multiplication of scale or ratios. While different arguments will have different effects on people, your statement that in the problem

2 + 2 + 2 = 3 x 2

the 3 on the RHS acts as an adjective and the 2 on the RHS acts as a noun, really struck me. I get it! Very subtle! Bravo!

Now, you have read my essay, so this may make some sense to you:

I *know* that you can show that the x operator above is associative and commutative which leads to it being an invertible transform of additivity. However, you can also show that x distributes over + in repeated addition, which constrains the quantification to being a log so that x must be multiplication.

Now what I bet you can show, is that the symmetries of ratios also lead to multiplication. So I am willing to bet that both problems "repeated addition" and "scaling" are quantified by the same function but for different reasons.

I am going to look into this as that would be really cool!

I also wanted to say that your circle-square problem is mesmerizing and I think that it effectively highlights the fact that there are some unrecognized subtleties still lingering in the metaphors that lead to the mathematics that we use (see Hestenes' essay and mine).

Thank you for a very enjoyable and thought-provoking essay!

Kevin Knuth

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I am glad you had a wonderful time talking to my good buddy David: such an outstanding individual.

The hardest thing to grasp is that there really are serious fundamental foundational issues with mathematics. The circle square illustration is just one of thousands. Recognizing that there are issues, is only part the problem, finding ways to illustrate them and solutions to them are others.

What I tried to point out in this paper is that dropping units may seem to be a logical approach to a generalized mathematics. However, so much is tossed out, requiring weird machinations to make it work. There are three different types of numbers: ordinal, cardinal and relationship. Currently mathematics switches back and forth between these without any indication that it has done so.

To be able to clearly see the issues requires what I call a “mathematics without numbers”. What becomes obvious is that numbers do not encode quantitative information but rather structural information. This then opens a doorway into a world of thinking, reasoning and logical connections. I believe that if Russell, Whitehead, Frege, etc. had discovered this way of thinking about number, math would be vastly different today.

Rob

BTW: The difficulty between the three’s and two’s was I believe (although I can’ t find it) pointed out by Hermann Weyl, as a difficulty with set theory.

The hardest thing to grasp is that there really are serious fundamental foundational issues with mathematics. The circle square illustration is just one of thousands. Recognizing that there are issues, is only part the problem, finding ways to illustrate them and solutions to them are others.

What I tried to point out in this paper is that dropping units may seem to be a logical approach to a generalized mathematics. However, so much is tossed out, requiring weird machinations to make it work. There are three different types of numbers: ordinal, cardinal and relationship. Currently mathematics switches back and forth between these without any indication that it has done so.

To be able to clearly see the issues requires what I call a “mathematics without numbers”. What becomes obvious is that numbers do not encode quantitative information but rather structural information. This then opens a doorway into a world of thinking, reasoning and logical connections. I believe that if Russell, Whitehead, Frege, etc. had discovered this way of thinking about number, math would be vastly different today.

Rob

BTW: The difficulty between the three’s and two’s was I believe (although I can’ t find it) pointed out by Hermann Weyl, as a difficulty with set theory.

thanks for sharing..this piece by R MacDuff is truly interesting..and I enjoyed reading all these comments..actually..they helped me understand this topic a little bit more and I think now I'm ready to submit my essay to http://essaytyper.pro..once again many thanks

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thanks for sharing..this piece by R MacDuff is truly interesting..and I enjoyed reading all these comments..actually..they helped me understand this topic a little bit more and I think now I'm ready to submit my essay to http://essaytyper.pro ..once again many thanks

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thuốc tăng thời gian quan hệtất cả mọi chức năng sau điều trị bệnh đều về mức ổn định rồi mà tại sao vẫn “yếu”, nó gần như không rõ nguyên nhân vậy. Sau đó mình tìm đến Trung tâm Thừa kế và Ứng dụng Đông y Việt Nam

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That was a very informative paper. I have always wondered how we can integrate science and maths. The information in this paper will sure help me to complete my essay at www.usacademicwriters.com

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That was a very informative paper. I have always wondered how we can integrate science and maths. The information in this paper will sure help me to complete my essay at essay typer

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Three Golden Rules for Integrating Science and Math

As you think about ways to integrate science and math, keep these three ideas in mind:

Plan lessons that use two or more of the skills listed above. Work from the basic principle that both math and science are about problem-solving.

Make hands-on activities a priority. When kids work with manipulatives, they gain the concrete experience that is so important to concept development.

Make science and math personal for kids by introducing activities with anecdotes, questions, or demonstrations that involve kids' local environments, whether it's their homes or the lake on the edge of town.

ve may bay di singapore - ve may bay di thai lan -ve may bay di nhat - ve may bay di ha noi ve may bay di phu quoc

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As you think about ways to integrate science and math, keep these three ideas in mind:

Plan lessons that use two or more of the skills listed above. Work from the basic principle that both math and science are about problem-solving.

Make hands-on activities a priority. When kids work with manipulatives, they gain the concrete experience that is so important to concept development.

Make science and math personal for kids by introducing activities with anecdotes, questions, or demonstrations that involve kids' local environments, whether it's their homes or the lake on the edge of town.

ve may bay di singapore - ve may bay di thai lan -ve may bay di nhat - ve may bay di ha noi ve may bay di phu quoc

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