What is mathematics?

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What is mathematics?
What is mathematics?
This is actually a very deep question, it may appear simple, but in fact is not. When we first think of math, we think – 2 + 2 = 4, or some similar arithmetic equation. This is definitely the beginning of math as we know it today. There are a number ways to look at math and I hope we can parse them out here. However, before we can identify what math is, we must understand what and where language and meaning comes from.
At the early stages of human development – there was no language. Estimates today date language anywhere from 60,000 to 200,000 years ago. So, what is language? It is a symbolic communication between two or more people. It can be sign language, verbal, artistic and/or written. However, before language can develop, there must be some meaning in the intellect of the being that wishes to communicate. This is an important point – meaning precedes language.
So, what do we use language for? We use it to express our ideas feelings, situations and we also use it to question. The question part is the basis of philosophy. Philosophy is the science of inquiry. It is the first science because it was used first to ask fundamental questions like – why am I here – where did I come from – where am I going? It is the inquiry of anything and/or something that forms the basis of philosophy. We make that inquiry with language. However the mechanics of the inquiry is philosophy.
So, the development of language also involved the development of philosophy. First understanding or meaning came to early humans – then language and philosophy advanced. Language is the workhorse that philosophy road and/or rides to express meaning and acquire new meaning. Language and philosophy evolved together by using investigative questions of inquiry.
Language in the beginning was very simple, but over time became more and more complex. The reason philosophy is the first science is because it asked questions about the use of the language itself. Philosophy posed questions that required language to become more and more complex. Philosophy and language are two sides of the same coin. They each used each other to advance.
Think about the first caveman to use language. He pounded on his crest and made the proclamation – “Brute”. He named himself. This declaration was understood by his neighbor. Then brute pounded on the crest of his neighbor and grunted to signify the inquiry – “what is your name”? The neighbor gave himself a name of “Hairy”. This first question or inquiry is the start of philosophy. This is how philosophy started and developed with the use of language.
That is what philosophy is – the science of inquiry. Over time the questions got more and more complex. I am not suggesting now that asking a person their name is what we call philosophy. This example is to show how language began concurrent with philosophy. Language in the beginning stated very simple things and asked very simple questions.
Language helped cognitive strength develop to the point that a person a little more advanced than Brute could pose the statement – “Brute & Hairy go hunting”. This is the beginning of mathematics. It is just a statement, not quite an equation – 1 + 1. It took time – maybe tens of thousands of years – before Brute got a full equation of 1 + 1 = 2. The point is that Brute plus Hairy is a symbolic statement of 1 + 1. Language itself is symbolic – the name Brute is a symbol of the person.
Philosophy is a symbolic inquiry into the nature of many things – so is mathematics. Philosophy and language were first. Math grew out of language and philosophical development. Math is a science of inquiry. In the same regard agriculture and biology grew from investigative inquiry. The first investigative inquiry was philosophy. Math came along somewhere down the road and used the same investigative inquiry methodology.
This is actually a very deep question, it may appear simple, but in fact is not. When we first think of math, we think – 2 + 2 = 4, or some similar arithmetic equation. This is definitely the beginning of math as we know it today. There are a number ways to look at math and I hope we can parse them out here. However, before we can identify what math is, we must understand what and where language and meaning comes from.
At the early stages of human development – there was no language. Estimates today date language anywhere from 60,000 to 200,000 years ago. So, what is language? It is a symbolic communication between two or more people. It can be sign language, verbal, artistic and/or written. However, before language can develop, there must be some meaning in the intellect of the being that wishes to communicate. This is an important point – meaning precedes language.
So, what do we use language for? We use it to express our ideas feelings, situations and we also use it to question. The question part is the basis of philosophy. Philosophy is the science of inquiry. It is the first science because it was used first to ask fundamental questions like – why am I here – where did I come from – where am I going? It is the inquiry of anything and/or something that forms the basis of philosophy. We make that inquiry with language. However the mechanics of the inquiry is philosophy.
So, the development of language also involved the development of philosophy. First understanding or meaning came to early humans – then language and philosophy advanced. Language is the workhorse that philosophy road and/or rides to express meaning and acquire new meaning. Language and philosophy evolved together by using investigative questions of inquiry.
Language in the beginning was very simple, but over time became more and more complex. The reason philosophy is the first science is because it asked questions about the use of the language itself. Philosophy posed questions that required language to become more and more complex. Philosophy and language are two sides of the same coin. They each used each other to advance.
Think about the first caveman to use language. He pounded on his crest and made the proclamation – “Brute”. He named himself. This declaration was understood by his neighbor. Then brute pounded on the crest of his neighbor and grunted to signify the inquiry – “what is your name”? The neighbor gave himself a name of “Hairy”. This first question or inquiry is the start of philosophy. This is how philosophy started and developed with the use of language.
That is what philosophy is – the science of inquiry. Over time the questions got more and more complex. I am not suggesting now that asking a person their name is what we call philosophy. This example is to show how language began concurrent with philosophy. Language in the beginning stated very simple things and asked very simple questions.
Language helped cognitive strength develop to the point that a person a little more advanced than Brute could pose the statement – “Brute & Hairy go hunting”. This is the beginning of mathematics. It is just a statement, not quite an equation – 1 + 1. It took time – maybe tens of thousands of years – before Brute got a full equation of 1 + 1 = 2. The point is that Brute plus Hairy is a symbolic statement of 1 + 1. Language itself is symbolic – the name Brute is a symbol of the person.
Philosophy is a symbolic inquiry into the nature of many things – so is mathematics. Philosophy and language were first. Math grew out of language and philosophical development. Math is a science of inquiry. In the same regard agriculture and biology grew from investigative inquiry. The first investigative inquiry was philosophy. Math came along somewhere down the road and used the same investigative inquiry methodology.

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Re: What is mathematics?
Philosophy is not a science at all. When Socrates asked the question, "How should I live?" how is that a question that science can answer? It can't. How is it a question that can be answered by referencing any amount of empirical evidence? It can't. That's what makes it a philosophical question. Another example of a philosophical question is "What is justice?" That also cannot be answered by any amount of empirical evidence. If a question can be answered by logic, math, science, or by empirical evidence, then it is not a philosophical question.
Mathematics is also not a science. It is nonempirical. Every mathematical object is an abstraction. Take the number one for example. No one can hold the number one in their hand, though, perhaps they can hold a symbol for the number one, but the numeral representing the number one is not the number itself. The number one exists no where in the external universe outside of our subjective minds that have come up with the abstraction of the number one. No one can ever do any physical experiments on abstractions, which means math is not a science, and never can be one. In math, we rely upon deductive reasoning to generate proofs, which then give us theorems we can use. There is no such thing as deductive proofs in an empirical science, like in physics, which proceeds by induction, drawing inferences from evidence.
You should also be aware that in grade schools, mathematics is seldom taught. Mathematics is not calculation, it is more than that. Mathematics basically deals with logical proofs. However, as a quirk of history, mathematical proofs are seldom taught at the grade school level, and not even during the math curriculum that forms the basis for an engineering and physics major. Real math typically begins after such classes, and the student usually gets initiated into real math in a class involving a subject like real analysis, discrete mathematics, abstract algebra, mathematical logic, or a transition course from computational courses and those that employ proofs. This is why so many people are confused on what math is  it is taught late in a math program, typically starting at some junior or senior level course, and seldom before then.
Mathematics is also not a science. It is nonempirical. Every mathematical object is an abstraction. Take the number one for example. No one can hold the number one in their hand, though, perhaps they can hold a symbol for the number one, but the numeral representing the number one is not the number itself. The number one exists no where in the external universe outside of our subjective minds that have come up with the abstraction of the number one. No one can ever do any physical experiments on abstractions, which means math is not a science, and never can be one. In math, we rely upon deductive reasoning to generate proofs, which then give us theorems we can use. There is no such thing as deductive proofs in an empirical science, like in physics, which proceeds by induction, drawing inferences from evidence.
You should also be aware that in grade schools, mathematics is seldom taught. Mathematics is not calculation, it is more than that. Mathematics basically deals with logical proofs. However, as a quirk of history, mathematical proofs are seldom taught at the grade school level, and not even during the math curriculum that forms the basis for an engineering and physics major. Real math typically begins after such classes, and the student usually gets initiated into real math in a class involving a subject like real analysis, discrete mathematics, abstract algebra, mathematical logic, or a transition course from computational courses and those that employ proofs. This is why so many people are confused on what math is  it is taught late in a math program, typically starting at some junior or senior level course, and seldom before then.

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Re: What is mathematics?
Fan of Science:
Geometry means literally a measure of the earth or land. It was, so to speak, grounded in the empirical practice of measuring and dividing the land. It began as an empirical science. Note that “science” is being used here to mean knowledge  the Latin scientia, which translates the Greek episteme. Again, geometric drawings were not simply heuristics, they served as the empirical objects from which geometric principles were discovered.
The ontology of mathematical objects remains an open question. Certainly modern mathematical language is a language of abstraction but this does not settle the question. There are still today mathematical Platonists  those, including professional mathematicians who hold that mathematical objects are not constructs but exist independently of the human mind.
Jacob Klein’s classic Greek Mathematical Thought and the Origin of Algebra is instructive. The Greek term “aristmos” does not mean the same thing as the modern concept of “number”. They differ in their intentionality. The Greek intends things, rather than the abstract concept of quantity. For the Greeks “one” is not a number. A number is two or more “ones” or units. The one or unit identifies what is to be counted. A number tells us how many of whatever it is that is being counted. What educators call “manipulables” are not just heuristics, they are the foundation of arithmetic in the Greek sense. It was with the move to the language of mathematical symbolism that mathematical objects became an abstraction.Mathematics is also not a science. It is nonempirical. Every mathematical object is an abstraction. Take the number one for example.
Geometry means literally a measure of the earth or land. It was, so to speak, grounded in the empirical practice of measuring and dividing the land. It began as an empirical science. Note that “science” is being used here to mean knowledge  the Latin scientia, which translates the Greek episteme. Again, geometric drawings were not simply heuristics, they served as the empirical objects from which geometric principles were discovered.
The ontology of mathematical objects remains an open question. Certainly modern mathematical language is a language of abstraction but this does not settle the question. There are still today mathematical Platonists  those, including professional mathematicians who hold that mathematical objects are not constructs but exist independently of the human mind.

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Re: What is mathematics?
The OP representation of language and humanity is plain ridiculous.
Anything Fan of Science says is on point.
Fool 
Origins are origins. In that sense we can ask searching questions about humanities capacity to understand the world. Yes, many do assume there is something more "real" to mathematical abstracts. They do say we can experience them in any true sense though. Just like we cannot experience "language", other than by its use.
There is most certainly a tangible relationship between mathematical theorem and reality.
Well, not really. The "objects" don't physically exist so you've kind of fallen over yourself there. We call a rabbit a rabbit, but there is no "absolute" rabbitness. With a circle it is ideal and formulated.
Mathematics is a language based on universals not variables. There can be no opinion on what a circle is because of this, whereas there can be differing opinions about things like "language" or "colour" due to personal experience and perspective.
All that said, it is quite obvious that without a physical reality to experience mathematics has no application. Maybe the universe is mathematical, maybe not. This is a matter of opinion and belief not a proof. Maybe the confusion presented here is all about the Platonic ideals?
Anything Fan of Science says is on point.
Fool 
Origins are origins. In that sense we can ask searching questions about humanities capacity to understand the world. Yes, many do assume there is something more "real" to mathematical abstracts. They do say we can experience them in any true sense though. Just like we cannot experience "language", other than by its use.
There is most certainly a tangible relationship between mathematical theorem and reality.
It was with the move to the language of mathematical symbolism that mathematical objects became an abstraction.
Well, not really. The "objects" don't physically exist so you've kind of fallen over yourself there. We call a rabbit a rabbit, but there is no "absolute" rabbitness. With a circle it is ideal and formulated.
Mathematics is a language based on universals not variables. There can be no opinion on what a circle is because of this, whereas there can be differing opinions about things like "language" or "colour" due to personal experience and perspective.
All that said, it is quite obvious that without a physical reality to experience mathematics has no application. Maybe the universe is mathematical, maybe not. This is a matter of opinion and belief not a proof. Maybe the confusion presented here is all about the Platonic ideals?
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Re: What is mathematics?
Nah, most of the time yours views and those of "Fan of Science" lack depth, i.e. focusing on the surface and not been able to dive deeper.Burning ghost wrote:The OP representation of language and humanity is plain ridiculous.
Anything Fan of Science says is on point.
Note this two images, of graphite pencil lead and diamonds
In your superficial approach based on common sense both of you will insist both pencil and diamond are not the same.
Yes, both are not the same because they do not look the same, they have different colors, density, hardness, etc.
BUT if we look on the basis of their atomic properties, they are exactly the same! Both graphite lead pencil and diamond are pure carbon  C  albeit of different density.
It is the same when both of you are arguing for Mathematics at the superficial levels, i.e. the various forms.
But what both of you are not able is to dig deeper to look at the 'atomic' or 'energy' level of mathematics.
At the 'atomic' and 'QM' levels [in comparison] of Mathematics is influenced by the finer empirical elements.
Mathematics cannot stand alone as absolute universals. Mathematics is influenced by experience either at the individual level or collective of the past [via DNA] and present.
Note even logic and reason which are the basis of Mathematics has their roots in biology, i.e. the brain and mind.
 The Evolution of Reason: Logic as a Branch of Biology (Cambridge Studies in Philosophy and Biology): William S. Cooper
http://assets.cambridge.org/052179/1960 ... 1960WS.pdf
Truth is never absolute to one perspective.
To understand the truths we must bring in a range of perspective [Nietzsche], i.e.
 1. common sense
2. Scientific
3. from molecular level to atomic to quantum
4. Philosphical perspective
5. Etc. etc.
Dive!!!
 Updated Fri Jun 30, 2017 11:02 pm to add the following 
@Woodart
I agree Philosophy and Science precede Mathematics.Woodart wrote:Philosophy is a symbolic inquiry into the nature of many things – so is mathematics. Philosophy and language were first. Math grew out of language and philosophical development. Math is a science of inquiry. In the same regard agriculture and biology grew from investigative inquiry. The first investigative inquiry was philosophy. Math came along somewhere down the road and used the same investigative inquiry methodology.
The problem with both BurningGhost and FanofScience is they do not dig deeper and view subjects in their full range of perspectives.
Etymologically 'science' meant 'to know'.
From Middle English science, scyence, a borrowing from Old French science, escience, from Latin scientia (“knowledge”), from sciens, the present participle stem of scire (“to know”).
https://en.wiktionary.org/wiki/science
Modern Science is still 'to know' but only within an agreed Framework and System.
Whilst Modern Science is somewhat regulated by a scientific community, we must break away from it when discussing 'Science' from a philosophical [as in this forum] perspective.
Mathematics is merely one form of 'to know' i.e. science at basic root level.
Thus at the 'atomic' perspective science definitely precede Mathematics.
Philosophy has its full range from atomic [substance] level to its various forms [Western, Eastern, academic, etc.].
At the deepest level of philosophy, it is the 'drive' of 'want to know.'
Thus the 'drive' and 'want/need' [philosophy] precede the 'to know' [science].
What is Mathematics?
In terms of origin it is
(1) Philosophy first [drive to know], then
(2) Science [how to know] and then
(3) Mathematics [knowing about quantity (numbers), structure, space, and change.].
Note whilst the ultimate level of philosophy is [drive to know], philosophy at its level of form is also a method of 'how to know.'
Notatheist. Religion is a critical necessity for humanity now, but not the FUTURE.

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Re: What is mathematics?
Math to my way thinking is fundamentally a type of counting. Of course it becomes much more abstract and complex today, but that is not how it started. Somewhere in the development of humans we began to count – things. Ask yourself – did we count before we had language? Maybe, I don’t know. Can we know? I don’t know that either. However, what I do know – or think – is that before the counting could be more complex – we needed language. Counting itself is a type of language. It could be argued that counting and language developed together – we don’t know.
Complex counting needs language. Therefore, I posit that language preceded what we think of elemental math (arithmetic). What is time? Time is a type of counting – a type of math. When did humans first become aware of time? A long time ago – before the Greeks – Chinese – before any civilization as we know it today. Humans were probably aware of time before the first human migration from Africa – 60,000 years ago. How can we know this as fact? Ask yourself – who thought of time first? It was a woman because she had a menstrual cycle. A woman is a walking clock. Women know time – they feel time. Men – not so much.
What else did early human women know about time? They knew their menstrual rhythm synced with cycle of the moon  28 days. So, not only are women walking clocks – they are our first astronomers. Women are responsible for the science of astronomy. Astronomy is like most other sciences because it is a composite. It has math – the heavens – time and biology. Math seems to be involved in every science. Do you know a science that does not use math? I cannot think of one – can you?
Complex counting needs language. Therefore, I posit that language preceded what we think of elemental math (arithmetic). What is time? Time is a type of counting – a type of math. When did humans first become aware of time? A long time ago – before the Greeks – Chinese – before any civilization as we know it today. Humans were probably aware of time before the first human migration from Africa – 60,000 years ago. How can we know this as fact? Ask yourself – who thought of time first? It was a woman because she had a menstrual cycle. A woman is a walking clock. Women know time – they feel time. Men – not so much.
What else did early human women know about time? They knew their menstrual rhythm synced with cycle of the moon  28 days. So, not only are women walking clocks – they are our first astronomers. Women are responsible for the science of astronomy. Astronomy is like most other sciences because it is a composite. It has math – the heavens – time and biology. Math seems to be involved in every science. Do you know a science that does not use math? I cannot think of one – can you?

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Re: What is mathematics?
Here is how Mathematics fit into the scheme of knowledge,
What is Mathematics?
In terms of origin it is
(1) Philosophy first [drive/need to know], then
(2) Science [how to know] and then
(3) Mathematics [that which is learnt  what one gets to know] [knowing about quantity (numbers), structure, space, and change.].
ThusThe word mathematics comes from the Greek μάθημα (máthēma), which, in the ancient Greek language, means "that which is learnt",[23] "what one gets to know" ......
wiki
What is Mathematics?
In terms of origin it is
(1) Philosophy first [drive/need to know], then
(2) Science [how to know] and then
(3) Mathematics [that which is learnt  what one gets to know] [knowing about quantity (numbers), structure, space, and change.].
Notatheist. Religion is a critical necessity for humanity now, but not the FUTURE.

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Re: What is mathematics?
I agree with Husserl here:
Spectrum 
note: no use of picture of Husserl's beard for effect.In any case, we can now recognize from all this that historicism, which wishes to clarify the historical or epistemological essence of mathematics from the standpoint of the magical circumstances or other manners of apperception of a timebound civilization, is mistaken in principle. For romantic spirits the mythologicalmagical elements of the historical and prehistorical aspects of mathematics may be particularly attractive; but to cling to this merely historically factual aspect of mathematics is precisely to lose oneself to a sort of romanticism and to overlook the genuine problem, the internalhistorical problem, the epistemological problem.  Edmund Husserl (The Origins of Geometry)
Spectrum 
I would never deny this. I would openly protest against anyone that did!It seem your views are a rigid oneperspective view
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Re: What is mathematics?
Burning Ghost:
First, you miss the point about intentionality. If a mathematical language refers to objects  a number of apples or oranges we cannot say that fruit does not exist or that any object that is counted does not exist, that there is no number of things or objects. Modern mathematical language, however, does not refer to particulars. This represents a fundamental change in the concept of number that we have difficulty recognizing because we have been trained to think in terms of mathematical abstraction.This is a conceptual shift in what numbers mean. To use Wittgenstein’s terminology they are two different language games.
Second, the assumption that only physical objects exist is questionable. The laws of nature are not physical objects but many physicists think they exist, that they are prescriptive and not simply descriptive.
Third, mathematical Platonists believe that mathematical objects, although not physical objects, do nonetheless exist.
From another post on Husserl’s prescientific attitude:
Jacob Klein’s “Phenomenology and the History of Science" reprinted here:issuu.com/bouvard6/docs/jacob_klein__h ... e insight.
What is at issue is the "sedimentation" of mathematics that occurs with the shift from the Greek concept of number in arithmetic and geometry, which was always tied to the question of “how many”, to mathematical symbolism as the “main instrument and real basis of mathematical physics” (10) .
“Universal science” as conceived by Descartes and others is an analytical art (ars analytice) that deals with symbols and the rules governing symbolic operations.
The prescientific attitude is one in which such assumptions do not come into play. Neither thought nor the world is mathematized in the sense of being regarded as if they were in truth the function of symbolic operations. It is historically prescientific not only in so far as it is temporally prior, it is prescientific in the sense of intentional history. It is in that sense an attitude that can be recovered.
Well, not really. The "objects" don't physically exist so you've kind of fallen over yourself there.
First, you miss the point about intentionality. If a mathematical language refers to objects  a number of apples or oranges we cannot say that fruit does not exist or that any object that is counted does not exist, that there is no number of things or objects. Modern mathematical language, however, does not refer to particulars. This represents a fundamental change in the concept of number that we have difficulty recognizing because we have been trained to think in terms of mathematical abstraction.This is a conceptual shift in what numbers mean. To use Wittgenstein’s terminology they are two different language games.
Second, the assumption that only physical objects exist is questionable. The laws of nature are not physical objects but many physicists think they exist, that they are prescriptive and not simply descriptive.
Third, mathematical Platonists believe that mathematical objects, although not physical objects, do nonetheless exist.
Jacob Klein was a student of Husserl. The quote from Husserl refers to the problem of historicism. Heidegger, Klein, and others look to history precisely to overcome the notion of historicism. His point is that the essence of mathematics is not a matter of contingent history.I agree with Husserl here:
From another post on Husserl’s prescientific attitude:
Jacob Klein’s “Phenomenology and the History of Science" reprinted here:issuu.com/bouvard6/docs/jacob_klein__h ... e insight.
What is at issue is the "sedimentation" of mathematics that occurs with the shift from the Greek concept of number in arithmetic and geometry, which was always tied to the question of “how many”, to mathematical symbolism as the “main instrument and real basis of mathematical physics” (10) .
“Universal science” as conceived by Descartes and others is an analytical art (ars analytice) that deals with symbols and the rules governing symbolic operations.
Upon the combined “sediments” reposes finally our actual interpretation of the world … the “scientific” attitude permeates all our thoughts and attitudes … We take for granted that there is a “true world” as revealed by the combined efforts of the scientists … This idea of a true, mathematically shaped world behind the “sensible” world, as a complex of mere appearances, determines also the scope of modern philosophy. We take the appearances of things as a kind of disguise concealing their true mathematical nature. (2021).
The prescientific attitude is one in which such assumptions do not come into play. Neither thought nor the world is mathematized in the sense of being regarded as if they were in truth the function of symbolic operations. It is historically prescientific not only in so far as it is temporally prior, it is prescientific in the sense of intentional history. It is in that sense an attitude that can be recovered.

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Re: What is mathematics?
Okay fool, the distinction then is that geometry is rigidly formulated? A circle is always teh same circle framed in a mathematical world.
When comes to the crunch I tend to parrot what Penrose says about mathematics, physics and consciousness as being different in term sof reality yet obviously connected.
From what little I knwo about mathematics I hav enever heard of it regarded as "the most empirical".
Who was it that said language is to thought what math is to phsyics? The main difference being that mathemtical language consists of "universal" terms only and is not open to mere opinions. Mathematics may bare relation to reality (in the physical sense) but as an abstract there is no need for this.
I am pretty sure this is what fan of science is rightly defending.
I take it you have come to a better understanding of Husserl now? Please share! I made the mistake in other thread of saying "logical investigations" when I meant Wittgensteins "Philospohical Investigations". I would liek to read Husserls work on this sometimes though.
When comes to the crunch I tend to parrot what Penrose says about mathematics, physics and consciousness as being different in term sof reality yet obviously connected.
From what little I knwo about mathematics I hav enever heard of it regarded as "the most empirical".
Who was it that said language is to thought what math is to phsyics? The main difference being that mathemtical language consists of "universal" terms only and is not open to mere opinions. Mathematics may bare relation to reality (in the physical sense) but as an abstract there is no need for this.
I am pretty sure this is what fan of science is rightly defending.
I take it you have come to a better understanding of Husserl now? Please share! I made the mistake in other thread of saying "logical investigations" when I meant Wittgensteins "Philospohical Investigations". I would liek to read Husserls work on this sometimes though.
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Re: What is mathematics?
Burning ghost:
There is an ambiguity to the claim that mathematical objects are abstractions. Such a view is consistent with Platonism in that intelligible objects are not concrete, they are not physical objects existing in space and time, but, on the other hand, the claim that mathematical objects are abstractions can mean that they do not have objective existence, that they do not exist except as abstractions from particulars.
If we look at noneuclidean geometries we actually has the reverse of what occurred in preeuclidean geometry. Noneuclidean geometries were originally useless, interesting but without practical application. They later found practical application in relativity. So here we have an abstract formal system that has a relation to reality that was not derived from the experience of reality, that is, empirically. The significance of the fact that we can "do" mathematics without regard to the world would be quite different if there were not such a strong relationship. Just what this relationship is based on refers back to the question of what mathematics is.
Here is an interesting talk on the philosophy of mathematics, “What are Numbers”:
https://www.youtube.com/watch?v=xXD57a5BEO0
No clear answer is given. Once again, it remains an open question.
And here is one by Ray Monk on the relationship between philosophy and mathematics, “Intro to the Philosophy of Mathematics (Ray Monk)”:
https://www.youtube.com/watch?v=bqGXdh6zb2k
As to the question of what numbers are he says that it is one of those questions that the more you think about it the less clear it gets.
And the problem of formal symbolic symbolic systems, “Kurt Gödel & the Limits of Mathematics”:
https://www.youtube.com/watch?v=jDyra0RHXs
What distinction? With Euclid geometry became axiomatic, but it began as a practical science applicable to such things as surveying and architecture. The properties of a circle are the same for all circles.Okay fool, the distinction then is that geometry is rigidly formulated? A circle is always teh same circle framed in a mathematical world.
I don’t know. Bacon and other modern scientists referred to mathematics as the universal language of nature.Who was it that said language is to thought what math is to phsyics?
Rather than claiming that mathematics is not science as Fan of Science does it would be more accurate to say it is not an empirical science.Mathematics may bare relation to reality (in the physical sense) but as an abstract there is no need for this.
I am pretty sure this is what fan of science is rightly defending.
There is an ambiguity to the claim that mathematical objects are abstractions. Such a view is consistent with Platonism in that intelligible objects are not concrete, they are not physical objects existing in space and time, but, on the other hand, the claim that mathematical objects are abstractions can mean that they do not have objective existence, that they do not exist except as abstractions from particulars.
If we look at noneuclidean geometries we actually has the reverse of what occurred in preeuclidean geometry. Noneuclidean geometries were originally useless, interesting but without practical application. They later found practical application in relativity. So here we have an abstract formal system that has a relation to reality that was not derived from the experience of reality, that is, empirically. The significance of the fact that we can "do" mathematics without regard to the world would be quite different if there were not such a strong relationship. Just what this relationship is based on refers back to the question of what mathematics is.
Here is an interesting talk on the philosophy of mathematics, “What are Numbers”:
https://www.youtube.com/watch?v=xXD57a5BEO0
No clear answer is given. Once again, it remains an open question.
And here is one by Ray Monk on the relationship between philosophy and mathematics, “Intro to the Philosophy of Mathematics (Ray Monk)”:
https://www.youtube.com/watch?v=bqGXdh6zb2k
As to the question of what numbers are he says that it is one of those questions that the more you think about it the less clear it gets.
And the problem of formal symbolic symbolic systems, “Kurt Gödel & the Limits of Mathematics”:
https://www.youtube.com/watch?v=jDyra0RHXs
No new insights. What I posted above is from a response to a comment of yours about Husserl’s use of the term ‘prescientific man’ in the topic “What is Husserl's basic project with his phenomenology?” My grasp of Husserl is tenuous at best and so every time I address him I come away with a better or perhaps just altered understanding. As to Jacob Klein, I came to him long before I became familiar with Husserl,through his commentaries on Plato. I found his work on Greek mathematics helpful with regard to the question of the one and the many as well as related issues of the Forms (each of which is one), reason/dianoia (the rational which functions by way of ratios and comparisons), and intellection/noesis (which alone can grasp the singularity of Forms themselves without comparing them to anything else).I take it you have come to a better understanding of Husserl now? Please share!

 Posts: 172
 Joined: May 26th, 2017, 1:39 pm
Re: What is mathematics?
If you don't want to take my word for it that math is completely nonempirical, then perhaps you would take the word of professional mathematicians? And before someone claims this a fallacy involving an appeal to authority, think again. Experts who know a subject are to be referenced over lay opinions. It's only a fallacy when the person is not really an expert in a field or the expert gives no reason other than simply "take my word for it" that we have a fallacy of appealing to an authority. Otherwise, it is perfectly rational to accept the arguments of an expert in the field.
I'll give three quotes, all from books I own, to once again set the record straight on what math is. It's exactly what I previously stated. (Hint: if I know nothing of math, then why is it I own these books?)
From the book, "A Book of Abstract Algebra, 2d Ed., published by Dover, 2010. The author is Charles C. Pinter. From page 13:
"To perceive why the axiomatic method is truly central to mathematics, we must keep one thing in mind: mathematics by its nature is essentially 'abstract.' For example, in geometry straight lines are not stretched threads, but a concept obtained by disregarding all the properties of stretched threads except that of extending in one direction. Similarly, the concept of a geometric figure is the result of idealizing from all the properties of actual objects and retaining only their spatial relationships. Now, since the objects of mathematics are 'abstractions,' it stands to reason that we must acquire knowledge about them by logic and not by observation or experiment (for how can one experiment with an abstract thought?)."
From the book "Mathematics: A Discrete Introduction, 2d Ed." Published by Thomson Bookstore (2006), and authored by Edward Scheinerman. From Page 2:
"Mathematics exists only in people's minds. There is no such 'thing' as the number 6. You can draw the symbol for the number 6 on a piece of paper, but you can't physically hold a 6 in your hands. Numbers, like all other mathematical objects, are purely conceptual. Mathematical objects come into existence by definitions."
From page 1: "The cornerstones of mathematics are definition, theorem, and proof. 'Definitions' specify precisely the concepts in which we are interested, 'theorems' assert exactly what is true about these concepts, and 'proofs' irrefutably demonstrate the truth of these assertions."
From the book titled "Measurement" written by Paul Lockhart, published by Harvard University Press (2012),
Page 5:
"What is a math problem? To a mathematician, a problem is a probe  a test of mathematical reality to see how it behaves. It is our way of 'poking it with a stick' and seeing what happens. We have a piece of mathematical reality, which may be a configuration of shapes, a number pattern, or what have you, and we want to understand what makes it tick: what does it do and why does it do it? So we poke it  only not with our hands and not with a stick. We have to poke it with our minds."
There is absolutely nothing empirical about math. While math may be used to solve empirical questions, math itself is, and shall remain, as 100% nonempirical.
 Updated July 1st, 2017, 4:28 pm to add the following 
With respect to the claim that math is basically counting, this is demonstrably false. Matching is more fundamental than counting, and this is what allows us to deal with such things as infinities. For example, if I know every person is able to occupy a seat in a large auditorium, with no seats remaining empty, then without knowing the actual number of seats or people present, I know they are equal to each other. In this way if one can pair a member of one infinite set with that of another infinite set, then one can tell that the sets are equal, despite being unable to "count" the members of either set, since they are both infinite. In this way, matching is more fundamental than counting, so the claim that math is basically about counting is false. If it were true, we would be missing out on a great deal of transfinite math.
 Updated July 1st, 2017, 4:32 pm to add the following 
With respect to the claim that philosophy came before math, think again. Why did Plato's academy require a student to know math? Why did Plato think about forms? It's because math was the basis for philosophy, not the other way around. Why did the ancient Greeks have democracy? It's because they applied the ideas of math to politics. When mathematicians prove something, they are merely providing an argument. Using these same ideas from math, one can use logic to come up with political arguments. It's not at all the case that philosophy gave rise to math, it's that math gave rise to philosophy, and it really shows when discussing Plato's forms, and what knowledge he required of his students.
 Updated July 1st, 2017, 4:32 pm to add the following 
With respect to the claim that philosophy came before math, think again. Why did Plato's academy require a student to know math? Why did Plato think about forms? It's because math was the basis for philosophy, not the other way around. Why did the ancient Greeks have democracy? It's because they applied the ideas of math to politics. When mathematicians prove something, they are merely providing an argument. Using these same ideas from math, one can use logic to come up with political arguments. It's not at all the case that philosophy gave rise to math, it's that math gave rise to philosophy, and it really shows when discussing Plato's forms, and what knowledge he required of his students.
I'll give three quotes, all from books I own, to once again set the record straight on what math is. It's exactly what I previously stated. (Hint: if I know nothing of math, then why is it I own these books?)
From the book, "A Book of Abstract Algebra, 2d Ed., published by Dover, 2010. The author is Charles C. Pinter. From page 13:
"To perceive why the axiomatic method is truly central to mathematics, we must keep one thing in mind: mathematics by its nature is essentially 'abstract.' For example, in geometry straight lines are not stretched threads, but a concept obtained by disregarding all the properties of stretched threads except that of extending in one direction. Similarly, the concept of a geometric figure is the result of idealizing from all the properties of actual objects and retaining only their spatial relationships. Now, since the objects of mathematics are 'abstractions,' it stands to reason that we must acquire knowledge about them by logic and not by observation or experiment (for how can one experiment with an abstract thought?)."
From the book "Mathematics: A Discrete Introduction, 2d Ed." Published by Thomson Bookstore (2006), and authored by Edward Scheinerman. From Page 2:
"Mathematics exists only in people's minds. There is no such 'thing' as the number 6. You can draw the symbol for the number 6 on a piece of paper, but you can't physically hold a 6 in your hands. Numbers, like all other mathematical objects, are purely conceptual. Mathematical objects come into existence by definitions."
From page 1: "The cornerstones of mathematics are definition, theorem, and proof. 'Definitions' specify precisely the concepts in which we are interested, 'theorems' assert exactly what is true about these concepts, and 'proofs' irrefutably demonstrate the truth of these assertions."
From the book titled "Measurement" written by Paul Lockhart, published by Harvard University Press (2012),
Page 5:
"What is a math problem? To a mathematician, a problem is a probe  a test of mathematical reality to see how it behaves. It is our way of 'poking it with a stick' and seeing what happens. We have a piece of mathematical reality, which may be a configuration of shapes, a number pattern, or what have you, and we want to understand what makes it tick: what does it do and why does it do it? So we poke it  only not with our hands and not with a stick. We have to poke it with our minds."
There is absolutely nothing empirical about math. While math may be used to solve empirical questions, math itself is, and shall remain, as 100% nonempirical.
 Updated July 1st, 2017, 4:28 pm to add the following 
With respect to the claim that math is basically counting, this is demonstrably false. Matching is more fundamental than counting, and this is what allows us to deal with such things as infinities. For example, if I know every person is able to occupy a seat in a large auditorium, with no seats remaining empty, then without knowing the actual number of seats or people present, I know they are equal to each other. In this way if one can pair a member of one infinite set with that of another infinite set, then one can tell that the sets are equal, despite being unable to "count" the members of either set, since they are both infinite. In this way, matching is more fundamental than counting, so the claim that math is basically about counting is false. If it were true, we would be missing out on a great deal of transfinite math.
 Updated July 1st, 2017, 4:32 pm to add the following 
With respect to the claim that philosophy came before math, think again. Why did Plato's academy require a student to know math? Why did Plato think about forms? It's because math was the basis for philosophy, not the other way around. Why did the ancient Greeks have democracy? It's because they applied the ideas of math to politics. When mathematicians prove something, they are merely providing an argument. Using these same ideas from math, one can use logic to come up with political arguments. It's not at all the case that philosophy gave rise to math, it's that math gave rise to philosophy, and it really shows when discussing Plato's forms, and what knowledge he required of his students.
 Updated July 1st, 2017, 4:32 pm to add the following 
With respect to the claim that philosophy came before math, think again. Why did Plato's academy require a student to know math? Why did Plato think about forms? It's because math was the basis for philosophy, not the other way around. Why did the ancient Greeks have democracy? It's because they applied the ideas of math to politics. When mathematicians prove something, they are merely providing an argument. Using these same ideas from math, one can use logic to come up with political arguments. It's not at all the case that philosophy gave rise to math, it's that math gave rise to philosophy, and it really shows when discussing Plato's forms, and what knowledge he required of his students.

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 Posts: 2970
 Joined: February 28th, 2014, 4:50 pm
Re: What is mathematics?
Fan of Science:
Compare what Paul Lockhart says here: https://www.youtube.com/watch?v=V1gT2f3Fe44
with your claim that :
Lockhart uses terms such as:
“Beauty” “elegant reason poems” “art of mathematics”
In my last post I linked to one of the many discussions of Godel's incompleteness theorems. There is more to mathematics in its contemporary versions than logical proofs.
I am not denying that modern mathematics is nonempirical but rather pointing out that a) modern mathematics does not have an exclusive claim on what mathematics is, b) that the fact that it is nonempirical does not mean that it is not a science unless one’s definition of science is limited to the empirical , c) the ontology of abstract mathematical objects remains problematic, and d) the relationship between mathematical objects and the physical world remains problematic. These are not problems that are generally addressed in textbooks such as “A Book of Abstract Algebra”. They are, however, addressed by philosophers of mathematics and by mathematician who are interested in philosophy.If you don't want to take my word for it that math is completely nonempirical …
You are looking at this through the wrong end of the telescope. Ancient mathematicians dealt with real world problems first  questions such as how many of this or that, the distribution of goods, and the measure and proportion of physical things, not such things as infinities.With respect to the claim that math is basically counting, this is demonstrably false. Matching is more fundamental than counting, and this is what allows us to deal with such things as infinities.
Compare what Paul Lockhart says here: https://www.youtube.com/watch?v=V1gT2f3Fe44
with your claim that :
Mathematics basically deals with logical proofs.
Lockhart uses terms such as:
“Beauty” “elegant reason poems” “art of mathematics”
In my last post I linked to one of the many discussions of Godel's incompleteness theorems. There is more to mathematics in its contemporary versions than logical proofs.
 Greta
 Site Admin
 Posts: 6683
 Joined: December 16th, 2013, 9:05 pm
Re: What is mathematics?
BG, Penrose is amongst of my favourite physicists but in this case I think his love of mathematics has lead to a bias, and like any rigorous scientist he cheerfully admits that is possible. It's a lovely idea  massive mathematical possibilities that contain a small concentration of physical entities in which just a small concentration is conscious. However, I think here that our language in describing the systems of reality is conflated with the actual patterns of reality, which are capable of being mapped mathematically as they are of being described with words, rendered on a 3D printer, or even presented as an "aural picture" via echolocation by a dolphin or bat to its peers.
Maybe mathematics is the deeper, more precise language of our future AI overlords. However, maths has historically been as inadequate in the social domain as social language is in describing how to build a bridge. Different tools for different tasks  qualitative and quantitative information. The demarcation between the two is where a current societal fault line lies. That is, increasingly quantitative methods are being used to assess quality. Numbers in certain contexts evoke emotional reactions as surely as adjectives  "you won a fortune" vs "you won ten million dollars". Emotions and dreams are being mapped ever more reliably by neuroscience. Surveys are conducted and reporting summarises most of the qualitative information into qualitative overviews, eg. x percent believe that y is the most important issue while z percent believe that w is most important.
Just as different cultures developed different languages, as Fooloso noted, they also developed variant mathematical protocols. I wonder why a global system of mathematics developed before a global language did (if one ever develops)? The decimal system is now dominant and mainstream in all civilised cultures (hohoho) rather than other numerical systems such as imperial, binary, hex, or based on 3s, 4s, 5s, 9s etc.
Maybe mathematics is the deeper, more precise language of our future AI overlords. However, maths has historically been as inadequate in the social domain as social language is in describing how to build a bridge. Different tools for different tasks  qualitative and quantitative information. The demarcation between the two is where a current societal fault line lies. That is, increasingly quantitative methods are being used to assess quality. Numbers in certain contexts evoke emotional reactions as surely as adjectives  "you won a fortune" vs "you won ten million dollars". Emotions and dreams are being mapped ever more reliably by neuroscience. Surveys are conducted and reporting summarises most of the qualitative information into qualitative overviews, eg. x percent believe that y is the most important issue while z percent believe that w is most important.
Just as different cultures developed different languages, as Fooloso noted, they also developed variant mathematical protocols. I wonder why a global system of mathematics developed before a global language did (if one ever develops)? The decimal system is now dominant and mainstream in all civilised cultures (hohoho) rather than other numerical systems such as imperial, binary, hex, or based on 3s, 4s, 5s, 9s etc.

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 Joined: March 3rd, 2017, 1:49 pm
Re: What is mathematics?
There is almost a spiritual quality to math – the universal language – and a reverence for the discipline. Math has always been like the judge, jury and executioner. I wonder if our AI overloads will even keep us around – if we can produce AI – they may think – we can’t top them. Will AI produce better music and art than us – will they really comprehend love? Will AI have emotion at all? I don’t think it is too far off in the future for real AI – what do you think?Greta wrote:
Maybe mathematics is the deeper, more precise language of our future AI overlords.