Scientific method

[Piero]

Coming back to the OP, I also think that a thesis must be proved in order to be fully accepted as new scientific knowledge. And this proof can be done theoretically or experimentally, depending on the problem to solve. Considering this proof as optional would be a big mistake and would lead to confusions and consequent deductive errors. With this, I believe in the importance of mathematical models for explaining nature because these can formally add evaluable information which could not easily captured without them.

The problem here is not easy though.. How do we design these models? How can we test these models? How can we refine these models? How do we build them? The idea behind a model is that this summarises literature knowledge (what we already know and proved). Therefore we can make predictions by simulating these models. This prediction must be tested. It can reveal novel information which can be correct or showing that the original knowledge included in the model is somehow incomplete and new other data must be included.

The latter is an important problem.. How much do we know about the universe? I think I can say very little.. Therefore these models will be inherently 'sloppy', or more technically will have many unknown parameters. And what about the physical laws? How much we know about them? I would say very little here too..

From this picture, it emerges that these models are so far inherently inaccurate both in the structure and in the parameter values. This means that their predictions will be weak somehow and that with our current knowledge a lot of models and predictions, even in contradiction between them can be made. That's why we have so many models of the universe.

So now the question is: are these worth? Well, I would say so, because still a simulation, even if wrong, can challenge your thought and drive you to formalise a theory in a better way. Clearly though, new experimental data is needed to test these models, otherwise they only remain at the level of prediction.

Why did they propose to remove experimental testing and accept theoretical physics as it is? I would be tempted to say that if so, they can have more publications, and therefore more funds, sadly, but also this is science.

[Piero]

'Also this is science' nowadays where quantity seems to count more than quality..

I won't prove this, but offer a simpler suggestion: how many people are getting a PhD nowadays compared to the last years?

Numbers are already online and the plot approximates an exponential curve starting from the late nineties! :-o

[Martin Meredith]

Agreed that it is an interesting thread. Here's a few more thoughts...

The questions of consistency and universality of mathematics are different. Logicians at the start of the 20th C did a pretty good job of showing that mathematics is not necessarily consistent. But it can still be universally inconsistent everywhere… For instance, to go back to Russell's paradox: is the set of all sets a member of itself? It is hard to imagine any part of the universe where this is not a paradox. It is a universal paradox, I would assert (without proof, however).

I think also we have to distinguish between mathematics per se (the part that may or may not be true everywhere) and mathematical models, which are approximations to reality. Newton's theory of gravity is different in essence from let's say, number theory. Newton's theory is not a mathematical theory, it is a mathematical model of something in the universe. It is possible to construct mathematical theories which have nothing to do with anything in the universe at the time of their production. The fact that Newton's gravity turned out to be incorrect does not affect the mathematical edifice in any way whatsoever. What would be shocking would be if -- say -- another of Newton's contributions, the calculus, was flawed. Einstein used Newton's calculus amongst other tools to demolish Newton's gravity. 

Having said that, I don't think the universality of mathematics is something that can be disproved or proved. To continue the discussion we need to decide on where the burden of proof lies.  Is it enough to demonstrate one part of mathematics that is universal?

Frankly, I find it hard to see how mathematics is not universal, if only because it is constructed based on agreed definitions. For example, if as a precondition we accept the definition of a prime number, then it is hard to see that prime number theorems are not universal. And if we don't accept the precondition of definition of a prime number, then we're not talking about the same kind of object anyway so any conclusions we draw are irrelevant. For me, mathematics is universal by definition. A prime is a prime is a prime. Is your prime different from my prime? If so, our definitions are different and it reduces to the triviality of misunderstanding or miscommunication, but it doesn't affect the essence of number theory.

Martin 

[ollypenrice]

I know at least bit about language but not a lot about mathematics. It seems that this puts me in a minority on this thread! Here are my objections to the 'maths as language' hypothesis but, as I say, I'm not a maths person.

- Maths (I won't be removing the 's' to please the americans ) is a language with such a limited vocabulary that I can't see it as being a language at all. Its vocabulary seems to be limited to numbers.  When we want to apply those numbers to nature we need words. The speed of light is c or 300,000 km per second. Can we have the idea of the speed of light without the words the speed of light? (ou la vitesse de la lumière etc etc...) I don't see how we can. And if this is so then maths is so incomplete as a language that it cannot be considered one at all, let alone a universal one. We meet an alien an she says 17. And we say 17 what? We want a word!

- If we insist on reductionist thinking (shoehorning maths into the term language - and why do we do this?) then would it not be better to describe maths as a universal grammar rather than a univesal language?

- The simplest tests of language are the most instructive. How do you say (or, even more importantly think)  'I am going to the dentist's on Thursday at four in the afternoon' without

words? Can it be described in maths without the assignment of number to words?

Why call maths a language? Why not call it maths? 

Olly

[Piero]

Martin, I think your post is very interesting and it was a pleasure to read it. I agreed that we didn't distinguished between pure and applied mathematics and number theory seems a valid example of branch of pure mathematics which could be universal. I am not convinced though whether all fields of pure mathematics can be part of an universal language. An example of this are the topics of group, rings, fields etc in abstract algebra and group theory.

Whether I can agree with you that a number here and else where can be interpreted with the same meaning, and therefore number properties, the notion of a group is an elegant definition which simplifies a large number of theorems and opens to new pure mathematics, but its definition is human to me and not universal. I would wonder whether for a casual intelligent alien species the concerpt of group is obvious, as well as such an alien species may have other definitions of mathematical structures, possibly more powerful than the ones we know.

The case of applied mathematics and in particular physical mathematics, I guess is also related on the universality of our physical constants.

In any case, I am not saying that mathematics is not an universal language. I am just saying that my view is, lets say, generally agnostic.

Number and group theories.. Beautiful beautiful fields of mathematics. Wish I had more brain to get into them properly.

[Piero]

Olly, mathematics is not just numbers, but has also variables, functions, quantifiers etc..

You can describe the sentence you said about going to a dentist using a formal language. You need to specify sets and assign a semantic to the values in these sets. Then the action of going can be expressed as a function.

The problem is not the word, but the semantic of the word. You need to provide a formal interpretation of course, but this is the same for natural languages.

The sentence you used contains a set of purely quantitative information (time) and an action (go). You can transform this sentence in a formal language.

What you certainly cannot do with a formal language is expressing the feelings that people like us have when we see your beatiful images!

Universal does not mean complete for living, but means informally that it is the most compact (reduced) language that can be used for describing a formal concept universally independently of culture or anything else. As Martin pointed out, the concept of number is not only valid here but likely valid everywhere, meaning that if you have to count, you or an alien species would do in a similar manner here or elsewhere (note: i mean the meaning of numbers, not the way one counts. The base is irrelevant here as you can convert between bases).

[Piero]

Maybe better to state it more clearly. When we talked about mathematics as universal language (I hope I can speak for everyone here) we meant mathematics as the universal tool for formalising nature.

If this were completely true it would mean that our mathematical and physical knowledge is the same everywhere in the universe.

If this is partially true (e.g number theory, but not the rest) it would mean that some of our mathematics is exactly the same everywhere, whereas the remaining parts are not.

Otherwise, it means that our mathematics is local or valid just for us.

I hope this clarifies part of this thread. Apologises, we should have stated this earlier, instead of assuming it.

Of course this opens to a interesting philosophical question: if it is only partially universal, why and what are the limits of our understanding of the universe?

Tricky to answer as we should know what is universal as Martin said previously. Even so, as Rob said, we won't be able to know the truth for what we are not sure is universal..

Hence, it seems to me a problem which is at most semi-decidible.. Another problem that we cannot really test it comprehensively..

[Piero]

Correction in the second IF: 'If this is partially true (e.g number theory, but not the rest) it would mean that some of our mathematics is exactly the same everywhere, whereas the remaining parts are not necessarily'.

[ollypenrice]

Thanks, Piero, that was very helpful and I have always believed in 'mathematics as the universal tool for formalising nature. '  I have no argument with that but have often wondered why this should be so. I suppose that it follows from the fact that nature is systematic, as it needs to be for it to be sufficiently coherent to produce atoms, molecules, predicable interactions and theorists wanting to formalise it. But what still bugs me about this is the suspicion that there might be some lurking tautology hiding in the shadows. I can't put my finger on it. 

Olly

[andrew s]

Thanks, Piero, that was very helpful and I have always believed in 'mathematics as the universal tool for formalising nature. '  I have no argument with that but have often wondered why this should be so. I suppose that it follows from the fact that nature is systematic, as it needs to be for it to be sufficiently coherent to produce atoms, molecules, predicable interactions and theorists wanting to formalise it. But what still bugs me about this is the suspicion that there might be some lurking tautology hiding in the shadows. I can't put my finger on it. 

Olly

I agree that we use Mathematics to ground our models of nature. However, it is notable that the models and mathematics we chose are often biased by the underlying "technology" of the time. We had the mechanical model of the universe deriving from the Victorian era and now the information centric theories in our information age.

For me Nature is what it is and our models seek to describe them. However, you need more than mathematics you need constraints from nature itself (boundary and other conditions). This for me separates Physics et.al. from pure mathematics and its ability to predict observables in the world is what separates it from meta-physics and sophistry in general.

There is no guarantee that different models are compatible with each other or that nature has a theory of everything. My hunch is that space time is continuous even at the Plank scale and below and so ultimately not describable by quantum mechanics. I feel this as I am unaware of an accepted Quantum formalism that does not use a continuous space time, the lack of evidence of a quantum foam from observation of photon flight times in cosmology and the failure to find a way to reconcile General Relativity and Quantum Mechanics. I don't in this assure either GR or QM are the correct theories in there domains. I look forward to being proved wrong.

Regards Andrew

[Tiki]

I know at least bit about language but not a lot about mathematics. It seems that this puts me in a minority on this thread! Here are my objections to the 'maths as language' hypothesis but, as I say, I'm not a maths person.

 

- Maths (I won't be removing the 's' to please the americans ) is a language with such a limited vocabulary that I can't see it as being a language at all. Its vocabulary seems to be limited to numbers.  When we want to apply those numbers to nature we need words. The speed of light is c or 300,000 km per second. Can we have the idea of the speed of light without the words the speed of light? (ou la vitesse de la lumière etc etc...) I don't see how we can. And if this is so then maths is so incomplete as a language that it cannot be considered one at all, let alone a universal one. We meet an alien an she says 17. And we say 17 what? We want a word!

 

- If we insist on reductionist thinking (shoehorning maths into the term language - and why do we do this?) then would it not be better to describe maths as a universal grammar rather than a univesal language?

 

- The simplest tests of language are the most instructive. How do you say (or, even more importantly think)  'I am going to the dentist's on Thursday at four in the afternoon' without

words? Can it be described in maths without the assignment of number to words?

 

Why call maths a language? Why not call it maths? 

 

Olly

If for instance you want to discuss physical science (chemisty, physics, astronomy) on a non-trivial level, then you are going to have to use an appropriate language. ie. maths. All other languages are less efficient. Unlike spoken/written languages, maths has it's own built in logic/rules and disaster ensues should you break any of them. It is possible to use bad grammar in English (for example) and not sacrifice meaning. Logically fallacious statements can also be grammatically correct in spoken/written language. I believe maths to be a language, it just that certain 'sentences' can be precluded. Maths can be descriptive which makes it more than just a grammar.

How does one describe an electron with words? Slithy tove perhaps:)

This post would perhaps make an English teacher wince but my old maths teacher would smile. Nearly 40 years ago as a twelve year old I had a maths lesson that would have even convinced Olly that maths is a language. The first lesson with Mr.Halder consisted of one question: What is mathematics? Despite haranguing us for an hour no-one was able to come up with a satisfactory answer. We were told to write 'Mathematics is a language' on the front of our exercise books.

[Martin Meredith]

I continue to insist that mathematics is a language given anything except the narrowest definition of a language e.g. 

http://www.thefreedictionary.com/language

"A system of signs used for communicating"

It isn't an accident that programming languages are called languages. And thanks to Turing we know that all programs are also clearly-defined mathematical functions. And given functional languages (e.g. LISP) there is no real difference between processes and data, so I can both represent your dental appointment and do computation on it (which bus do I need to catch to make it in time?).

Mathematics is more than a tool for formalising nature. I can't think of a single attribute of a language as conventionally understood that mathematics does not possess. Why do we have syntax? Essentially to allow us to build an infinite number of sentences from a finite number of building blocks. We don't have a single word for "dentists-appointment-at-two-thirty" because that would be crazily inefficient. The same is the case for mathematics but more so. From a small number of building blocks we can create a huge number of meaningful mathematical expressions. Indeed, mathematics is more efficient. With just two concepts (the empty set and set union) we can define the entire set of natural numbers. In a programming language with say, 50 keywords, and maybe a dozen operations, we can create the world wide web. (Actually, in practice, if we're prepared to operate at machine code level we don't even need this many).

So much for syntax; what about meaning?  Who cannot be amazed by the beauty of the innocuous-looking Euler's identity 

     e^i.pi + 1 = 0

where i is the square root of -1. Leaving aside the fact that it links three of the most common operators and five of the most most important quantities in maths in a way that is totally unexpected (why should weird constants like pi and e and even weirder things like i be linked in such a simple manner?), and leaving aside that there is some fundamental truth buried in there, no mathematician could look at this without seeing meaning. Some will see a circle, others will see sine waves (and hear music), others will see the building block for the transforms that underlie a lot of astronomical image processing, others will see links to electromagnetism, …. How much meaning can be packed into a single equation? What's more, monolingual native speakers of Turkish, Finnish, Hungarian, Basque, Cantonese, Thai, .. you name it, with no words in common, will see the beauty of this identity and in principle derive the same conclusions from it.

This mathematical identity expresses a universal truth in a language that is easily understood.

Martin

[ollypenrice]

By Gad, I do believe you're persuading me, Martin. 

But when you say, 'Some will see a circle, others will see sine waves (and hear music), others will see the building block for the transforms that underlie a lot of astronomical image processing, others will see links to electromagnetism,' how can you see these things (see in the metaphorical sense of 'conceive') without a different kind of language capable of identifying 'electromagnetism' or 'music' etc? This is returning to my point about the nature of vocabulary.

Olly

Edit. Or, to put it another way, isn't maths a language which is really only competent to describe relationships between things? (And here it is prodigiously competent.)

[symesie04]

'Also this is science' nowadays where quantity seems to count more than quality..

I won't prove this, but offer a simpler suggestion: how many people are getting a PhD nowadays compared to the last years?

Numbers are already online and the plot approximates an exponential curve starting from the late nineties! :-o

If i read this right you are suggesting that as more people are achieving PHD status then the quality must be of a lowered standard? If so have you not fallen foul of your own critique of acceptance without testing? I dont think there is any evidence that the increase in those acquiring a PHD is because of a decrease in standards. There are many possible reasons for the increase the first being the ever increasing numbers of those starting PHD courses (and i can supply proof of increases if required). Thats not even to mention a whole range of other possible criteria such as online access to research material, technology allowing improved revision techniques and time spent revising. And im sure there may be many other con-tributaries social awareness and expectancy for instance. Just a thought.

[Qualia]

This mathematical identity expresses a universal truth

Aye, maths is a great thing and there can be no serious disagreement about this. The entire edifice of modern science and technology bears witness to this fact.

But the question remains, - what exactly is being said of that "universal truth"? What ground or foundation is being used to confidently declare that mathematical statements are 'universal truths'? Or, again, what is the foundation of maths which enables its statements to be considered universally true?

Highlighting one of my concerns - for good or for bad - there appears to be the creeping in of Platonic sentiments, namely, that the world/universe is rooted in a mathematical realm. That in a very real way, "the book of the universe is written in the language of mathematics" and by inference, the semi-Abrahamic notion that we are "created in the image of God Math". A statement which would not only be begging the question but raising even greater problems, not least those faced by theologians and other religious or quasi-religious systems.

[symesie04]

I wonder if maths being the language of the universe is just sciences replacement for that that it so readily dismisses. But as fabulously interesting the debate about the implications of maths being a universal language has on our psychological beliefs is it seems to me that what is still essentially being debated here is in direct relation to the OP. Im pretty sure that we have to accept that testing the concept of maths being a universal language will always be out of our reach. We may receive more and more evidence to point to it as being correct but without exploring every nook and cranny of the universe we can never declare that it is upheld throughout the universe. Of course at this point in our evolution we cannot even be assured that maths is not purely mankinds interpretation of its environment therefore rendering it jibberish to an alien. But as we have no means by to test the theory then it all boils down again to the OP and do we accept a concept without testament to its validity? If not then what then does that mean for maths? If we do not accept maths as being a constant in the universe then has it any authority within our realm?

[ollypenrice]

Highlighting one of my concerns - for good or for bad - there appears to be the creeping in of Platonic sentiments, namely, that the world/universe is rooted in a mathematical realm. That in a very real way, "the book of the universe is written in the language of mathematics" and by inference, the semi-Abrahamic notion that we are "created in the image of God Math". A statement which would not only be begging the question but raising even greater problems, not least those faced by theologians and other religious or quasi-religious systems.

It's in this area that I suspect my tautology might appear, if it ever will. We model the universe in maths. It looks like maths. What else would it look like? I suppose it could refuse to be model-able in maths but it doesn't. Or maybe it does, of course, which is why we have no model of what it's really like... This assumes that it's meaningful to talk of a universe existing beyond our perception of it, which may be quite an assumption.

I must read a lot more about maths.

Olly

[Piero]

If i read this right you are suggesting that as more people are achieving PHD status then the quality must be of a lowered standard? If so have you not fallen foul of your own critique of acceptance without testing? I dont think there is any evidence that the increase in those acquiring a PHD is because of a decrease in standards. There are many possible reasons for the increase the first being the ever increasing numbers of those starting PHD courses (and i can supply proof of increases if required). Thats not even to mention a whole range of other possible criteria such as online access to research material, technology allowing improved revision techniques and time spent revising. And im sure there may be many other con-tributaries social awareness and expectancy for instance. Just a thought.

Thanks for your post.

The fact that there are more courses, social awareness, etc is not a problem to me. When I said that nowadays in science quantity seems more important than quality, I meant that: 

1. The more publications a project leader has per year, the better is. This was not the case in the past. 

2. Whereas a project leader in the past had few PhD students, nowadays you see groups that look like more an army! 

3. From point 1 and 2, it happens that these PhD students are pushed more at producing than at stimulating their own ideas and if they do not get a paper out, they will hardly find a good postdoc position.

4. From point 4, nowadays the number of journals is exploding and impact factor metric rules. There are so many articles out there than people reading them. So many articles are completely forgotten. In addition, how are these reviewed? There is not a commonly standard level of quality and process for papers to review, and the reason for this, to me, is that eventually people want their papers out, because this leads to new fundings. Now, is this science or business.. ? 

5. If the number of people doing a PhD increases and the standard of these were the same, then you would expect a consequent increase in the number of people failing their PhD at the first year.. Conversely, you find a decrease in this, meaning to me that most of PhD students simply pass through. You might argue that the course is developed better and that is the reason why less PhD student fail, although honestly I do not believe in this since a project leader has instead much less time to dedicate to his/her students nowadays also because these are too many (point 2). 

Now, although I have not formally proved this, but I think I made a reasonable argument for what I said.  To my view there is a lot of politics and business in academia and the system of control does not work properly. The reason why I think this is because the entire process is run and managed by the same people.

Regarding papers, if a principal investigator (PI) has a paper to submit, and s/he sends it to a journal where s/he knows the editor and suggests reviewers who s/he knows. 

If this paper goes through, it will be more likely that s/he will receive funds later. Who decides whether s/he can receive those fundings? Other PI, possibly people that s/he knows too..

And if s/he can, s/he becomes editor of her/his own journal and technically submits her/his papers to it too.. evading the quality process. 

Unfortunately, who is going to pay for this are PhD students in primis, as they believe in science, whereas instead they get stuck in a business machinery in most of the cases. 

[Piero]

It's in this area that I suspect my tautology might appear, if it ever will. We model the universe in maths. It looks like maths. What else would it look like? I suppose it could refuse to be model-able in maths but it doesn't. Or maybe it does, of course, which is why we have no model of what it's really like... This assumes that it's meaningful to talk of a universe existing beyond our perception of it, which may be quite an assumption.

I must read a lot more about maths.

Olly

I am deeply intrigued, but I am not sure I understood what you mean. Could you explain what you suppose "'I suppose it could ..." a bit more, please? Thanks!

[ollypenrice]

I am deeply intrigued, but I am not sure I understood what you mean. Could you explain what you suppose "'I suppose it could ..." a bit more, please? Thanks!

What I meant was that the universe might not have yeilded to mathematical modeling. Indeed, sometimes it has been very reluctant to do so.  Schrodinger, I think, was asked if he had any questions for god and replied (paraphrasing from memory), Yes, quantum mechanics and turbulent flow. I think I might get an answer on quantum mechanics. (I like to mention this to the many hang glider and paraglider pilots who live around here and who believe that their devices are essentially sound... ) 

Here's the thing; our thinking advances along lines of progress. Where we have advanced with specific tools we think further. Where, with the tools we have, we have failed to advance we give up. This is a self selecting process. It should sound a warning, no?

Olly

[Piero]

What I meant was that the universe might not have yeilded to mathematical modeling. Indeed, sometimes it has been very reluctant to do so.  Schrodinger, I think, was asked if he had any questions for god and replied (paraphrasing from memory), Yes, quantum mechanics and turbulent flow. I think I might get an answer on quantum mechanics. (I like to mention this to the many hang glider and paraglider pilots who live around here and who believe that their devices are essentially sound... ) 

Here's the thing; our thinking advances along lines of progress. Where we have advanced with specific tools we think further. Where, with the tools we have, we have failed to advance we give up. This is a self selecting process. It should sound a warning, no?

Olly

Regarding your second paragraph, I would say that people "step back temporarily". Discoveries can take different times to be elucidated and if a discovery is not found, it can mean that it is either too early or that it requires a more original thinking. Clearly you cannot know this a priory and that is why you need to invest some time (and money) doing some research. 

About the first paragraph instead, a mathematical model is not necessarily deterministic. In general, it is a way to define a mechanism including some properties/behaviour of a natural system. For instance, if the system is strongly regulated by stochastic events, these must be taken into account when building a mathematical model (and when simulated, will produce a stochastic simulation). 

The point is that a model is an abstraction, meaning that you are considering only a subset of all the laws governing the system, and that is why every model is inherently inaccurate (and that's why testing them is very important ). Despite this, it can be very informative even if simplified. Eventually it also depends on what you are looking at. 

To my mind two things are inside a model. These two things are: structure and parameters. 

1) structure: the entire set of natural laws and how these are related to each other

2) parameters: how you can, let's say, setup those previous laws. 

Structure and parameters are of course inter-dependent and the system is the combination of the two. 

Then there is an interesting function which is time, which plays on the system (or couple structure-parameters) as an evolution.

I wonder whether Nature has somehow a similarity with this, and our science is the detection of its two components (physical laws and constants). There are a few questions though (which are related to the universality of mathematics too):

a ) Are these components (structure + parameters) local to us or the same everywhere? 

b ) Assuming these are the same everywhere, is the configuration that we perceive the only possible one?

c ) How (and why) did this (or these?) structure and parameters originate? 

It seems that there isn't any other way in the universe to represents natural numbers expect for the way that we represent them. And therefore, assuming this, we can infer a potentially infinite number of properties over this set of numbers. 

But is this really true or it depends on our intrinsic way of thinking? What I mean is that we created these numbers, originally because we needed them for counting and then we continued studying them in their purity as well as we applied them to interprete nature in a formal and coincise way. However, could it be that there are one or more other ways to interprete nature without using our mathematics? and if so, are there forms more advanced and sophisticated of our mathematics?

Although this might seem an erratic thought, I think that it deserves notice to recognise that animals for instance comunicate between each others and this communication happens in a way well different from ours. The entire structure is different. In the same way, an alien society could comunicate in a much more advanced way than ours using a different structure. Therefore, why couldn't exist another way to formalise nature that is not our mathematics?

[Qualia]

Brilliant insights and reply, Piero. Thank you for sharing them.

I was cycling to work today and along the way reasoned that I have come to the limits of my reason regarding this topic. It's escaping my ability to intelligibly communicate or add anything of significance. I'm not versed enough in maths or the topic to do much else. 

However, the whiff of Kant in me sprung upon this and set my wondering:

I wonder whether Nature has somehow a similarity with this, and our science is the detection of its two components (physical laws and constants).

This is not addressed only to Piero. It would be nice to hear all ideas.

Do you think our maths is a detection of the underlying form or forms in Nature, that our maths works because it is literally derived from the physical world and exist independently of the human mind? Or, do you think the form or forms underlying the Physical World, the 'out-there', come into existence once they are constructed by the human mind operating in math-mode? Or, again, do you think there might be some middle ground where in some sense it is a combination of the two?

[andrew s]

This is not addressed only to Piero. It would be nice to hear all ideas.

Do you think our maths is a detection of the underlying form or forms in Nature, that our maths works because it is literally derived from the physical world and exist independently of the human mind? Or, do you think the form or forms underlying the Physical World, the 'out-there', come into existence once they are constructed by the human mind operating in math-mode? Or, again, do you think there might be some middle ground where in some sense it is a combination of the two?

These are issues that have been debated long and hard for time immemorial and as far as I am aware no strong consensus exists with scientist or non-scientists alike.

My two penny worth is that I believe that reality exists completely independent of my existence or the existence of anyone else. By this I mean it is not a construct of some mental states of intelligent observers.

On the other hand our concept (view) of reality is limited by our evolved senses and the technology we have developed to extend them and also by the structure and function of our brains and thus our abilities to think and conceptualise.

Man has used many approaches to seek an understanding of our reality (i.e. nature, the universe etc.) but to my mind the scientific method has proven the most effective. Not that the "scientific method" is a singular unchanging entity it has evolved and continues to do so and different field of science use different variants.

By effective I mean given us the most predictive and retrodictive ability and thus enabled us to shape and control the environment in which we find ourselves. It works!

As I said in an earlier post our theories are models we use to make senses of reality they are not reality itself, be they mathematical or not. They seem to apply equally well (or badly) across the observable universe as for example, we don't need different models of say the Hydrogen atom to explain emissions form stars at different distances.

As mathematics is a human construct, and its results are accepted ultimately by agreement that one step in a mathematical proof follows form the previous one, then one could imagine a set of beings that would not accept these conclusion. I think, however, that that is very unlikely for any beings we could communicate with.

Regards Andrew 

[ollypenrice]

Do you think our maths is a detection of the underlying form or forms in Nature, that our maths works because it is literally derived from the physical world and exist independently of the human mind?

This is a beast of a question! Had there not been things, would there have been number? Think of a universe not void (which I guess would not meaningfully be a universe?) but in which energy had never congealed into matter and was in a totally smooth distribution. Would there be number in such a universe? Only if we stuck our heads into it, I'd say!

But there are things we have stuck our heads into our universe and it does have number. So I have to conclude that, yes, maths is indeed a detection of the underlying forms of nature and that we should not, therefore, conclude that there is anything remarkable about the ability of maths to describe it so well. That's answered a question I've been pondering for some time so thanks for the question, Qualia.

What are the constraints on our thinking? There are many, of course - and many will remain unknowable. But one key constraint which we sometimes do think about is time, and we should spend more time on it... The example which I always give is of the photons lining up for the second part of the double slit experiment and being told that, this time, they'll be fired one at once. They stare back blankly and ask, 'What does one at once mean?'

Olly

[symesie04]

Crikey Qualia that really is a tin of worms opened there. Indeed is maths a sound reflection of natures underlying design or does humans limited dimensions of thought force us to shoe horn our perceptions into a mould that fits? A proverbial "round peg square hole" except perhaps the peg isnt quite round but certainly not square but just happens to seem to fit roughly, not perfect but seems to fit and therefore we accept it as a fair reflection of natures design. Of course it also begs the question could we ever recognize either way? If nature is not bound buy mathematics and instead by some other construct then would we ever be able to hope to recognize it even if dangled in front of our faces. With our whole lives so dependent on maths its hard to imagine how we could see it.