"I do not wish to judge how far my efforts coincide with those of other philosophers. Indeed, what I have written here makes no claim to novelty in detail, and the reason why I give no sources is that it is a matter of indifference to me whether the thoughts that I have had have been anticipated by someone else."
Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Philosophy Knols of the same writer
This Knol is part of a series of philosophy-related Knols I have written, which are listed below.
CONTRIBUTE in Philosophy VIA ART!
Do you want to express your philosophical thoughts via art? Do you think great philosophical ideas can only find their way through poetry? Feel free to enter the open Knol Poetry Contest!!!
I. Introduction - About Science
Exact Science has been for a long time now the tool used by most humans in order to understand the physical world. The great successes of sciences like physics or astronomy in predicting things and helping the everyday life of people has established "exact" science as the ultimate tool to discover the truth. This has unfortunately led to a decline of the importance other scientific fields of human thinking have - sectors as important as philosophy, history, religion, linguistics. We tend to forget that law or history can be as scientific as quantum physics. We tend to forget that science is seeking the truth with a systematic way and that fields of science like physics or chemistry are only a small portion of it. Science is not only physics. It is also the exploration of human affairs, the attempt to explain why humans fight each other, the struggle to understand what "bad and good" is. Anthropology, sociology and law are fields of science in the same way cosmology is. Science is not only about evidence. Many people confuse science with "empiricism", that is the philosophical dogma that all valid knowledge can be based on human experience only. These people tend to forget that even cosmologists and physicists dream about parallel worlds or alternate universes, hypotheses that can never be proved or disproved by evidence.
Before someone starts dealing with science, he/she must pay a visit to a good philosophical dictionary so as to make sure that he understands all words with their correct meaning. I strongly recommend the Dictionary of Philosophy, Dagobert D. Runes, 1942, New York as a basis for your knowledge seeking.
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| René Descartes. Portrait after Frans Hals, 1648. A great philosopher of science |
It is unfortunate that the abovementioned misconceptions have led humans away from thinking about humans. Fortunately many thinking people of our time have pinpointed the problem and insist on thinking "scientifically", rather than thinking "scientifically as physicists"...Science is not only "thinking numbers based on physical evidence" - it can also be "thinking logically".
II. About exact science: A definition
Exact science is based on observation and induction. You observe one phenomenon, then you observe a similar one, then another and so on, until you reach to a conclusion about how such phenomena work. Then you formulate a model that predicts similar phenomena in the future. If your model works, then your theory is accepted by other scientists (the infamous "peer-review" model, which can lead to weird applications of auto-cencorship for not widely accepted ideas [1] - but this is only out of the scope of this article). Until a phenomenon not following your set of rules is observed...
Other fields of science
Besides the "exact" sciences, there are many other fields which exist. These other scientific fields are based on the "logical and structured way of analyzing a specific subject" and include sectors like anthropology, linguistics, law, psychology etc. One of the most negative impact of the "exact" sciences' domination in modern times is the demise of these other equally "scientific" fields. Fortunately most people are beginning to realize the mistake in that and a regeneration of interest in these "human" sciences is observed.
Definition of Science
Science is the systematic search for truth. And in that context physics and chemistry are not the only fields of science that exist. Many people confuse science with "empiricism" (i.e. all knowledge is based on human experience only). Empiricism is a philosophical dogma and a very outdated one, I might say. [2] My articles have the purpose to clear such confusions.
III. Limitations of exact science
Exact sciences like physics have many limitations, often disregarded by their "followers" (i.e. people who think that measuring, evidence-based exact science is all that there is in the scetor of human knowledge-seeking). I will discuss the main of these limitations here.
1. Axioms of Exact Science turn into Dogmas
The main limitations of science are a result of the things science takes from granted - i.e. of the axioms it uses as the basis for any further research. When you use an axiom, then you start on the wrong foot. You cannot prove axioms and if you base everything on them, then you guarantee that your whole theory cannot be ultimately be proved.
Some of the main (wrong, according to my opinion) axioms of exact sciences are:
- Sciences like physics have excluded anything spiritual from the "equations" of the universe. Physics axiomatically says that the cosmos consists of nothing more than electrones, protons and other particles that obey some physical laws - while at the same time denying anything of 'spiritual' nature. In that context, no wander scientists cannot find 'evidence' for God or clues for the existence of any kind of 'Purpose' in our existence. After you have axiomatically accepted that a system consists of A, B and C only, how do you expect to find D anywhere? [2] The science of biology also makes the same mistake: it uses an axiom as dogma without ever questioning whether it is right or wrong. Modern biology takes for granted that humans are animals and that evolution rules apply to everything, without prooving it. Modern science is based on the doctrine of materialism [3], also known as "physicalism" [4]. However we must remember that materialism is a dogma and not a proven case. [5]
- All natural laws are universal and apply to everything and everyone. This has never been proven yet. It is just a thing scientists believe that is true. [6] This axiom is also what causes problems in the explanation of the basis of our human nature: "free will". If we are to accept the existence of universal physical laws then no free will can exist. But most of us think otherwise. Most of use feel that we "decide" what to do. Science simply cannot explain that.
- All things are measurable. This is based on a highly materialistic view of the world and has nothing to do with reality (whatever that word means...). Things like moral, emotions, aesthetics and love cannot be measured, even though they are very important for human life. Science fields like physics, mathematics and chemistry simply cannot deal with these things.
- We can know everything. Many scientists wrongly believe that humans will someday come to know everything, that there is no such thing as "un-knowable" area. However Godel has proved them wrong: he proved that there are truths that cannot be proved anytime by anyone.
- Our Logic works correctly. This may sound odd, but it was one of the main axioms used to form the foundations of science. Out logic "must" work properly if we are to trust it. [6]
- Everything can be replicated in an experiment. Because science needs experiments to prove or disprove theories, it cannot deal with one-off events (that happen only once and cannot be reproduced). This is a simple but rather important axiom: we believe that all phenomena can be studied by experiment or observation. What happens with things that happen only once in the Universe's history?
- Mediocrity and Copernican principle (principle = not proven declaration): In cosmology, the Copernican principle, named after Nicolaus Copernicus, states the Earth is not in a central, specially favoured position. More recently, the principle is generalised to the simple statement that humans are not privileged observers. In this sense, it is equivalent to the mediocrity principle, with significant implications in the philosophy of science. The mediocrity principle is the notion in philosophy of science that there is nothing special about humans or the Earth. These two principles combined form the basis of our current cosmology. [7] [8] Even though the Theory of Relativity states that we can choose ANY point of reference for basis of calculations in the universe (which means that we can easily put Earth in the center of everything and form the physical laws with that as a reference point), even though the universe seems the same in any direction we may look and all galaxies seem to drive away from us (as if...we were in the center of the universe as Hubble himself admitted, but then denied simply because that would be opposing these two "principles"! [9] [10] ) some people still maintain their "belief" (dogma?) that humans are just grains of dust in the cosmos. I do not say that I know for sure that we are or we are not. But I cannot agree with someone that takes that for granted. I cannot argue with dogmatic people...[read the related article Earth is at the Center of the Universe? for more on that]
The abovementioned axioms are the basis for the limitations of science. However one might say that this is not a problem - that every theory must start from somewhere. I cannot agree more. My objection is not in the use of axioms by itself, but in our completely forgetting that we use those axioms. When we believe that our science is based on "true" arguments, then we forget the basis of our science. When we forget that we use axioms (and that if we use other axioms we will reach completely different conclusions) then those axioms turn into dogmas. And dogmatism, in any form, is not a good thing...
2. Exact sciences cannot measure
It sounds strange to say "exact science cannot be exact", but it is absolutely true!
Ask a physicist to measure the length of a table in your house. He will take the necessary tool to measure length and will measure, let's say, 80 cm length. But is this true?
Just imagine another physicist comes to your house and measures the same table with a different - but more accurate - tool. He will measure the length equal to 80.095 cm. Is this correct?
Another ones comes to your house and with a tool of greater accuracy measures the length equal to 80.0949988988171716 cm. Could this be our final measurement?
I am afraid not. Every time you use greater accuracy, you end up with a different number!
Ok, one might say. I will use a SEM and measure the mength at the atomic level. Ok. Try it. You attempt to do just that only to realize that at an atomic level you stumble on the uncertainty principle of Heisenberg: you cannot know where the electrons are exactly!
What does this mean? Does it mean the table has no length? It must have a length, since we see it in front of our eyes. The point here is that the table has a length, but our science cannot measure it. That is not a limitation of our science today. We know that we will never be able to measure the exact length of the table due to quantum mechanic phenomena occuring.
Final outcome: you do not know and will never know the exact length of your kitchen table! Weird conclusion for what we call the "exact sciences"...
3. Main limitations of scientific tools
Other limitations of science come from the tools it uses. Some of them are discussed below.
1. The tool of 'induction' is by itself a problem for science. Lets say you observe a frog and see its green. Then another frog and you see that it is green also. Then another, and another - until you are convinced that all frogs are green. Then you write a theory about frogs. Everythings seems quite good up to this point, quite 'real', quite 'scientific'. Until you observe a black frog... [3]
2. Science is based on our senses. The limitations of them may pose significant limitations to how we understand the world, that we may never be aware of [1]. Since we do not know if our senses work "correctly" (mainly because we do not have a benchmark as to what is the "correct" way for senses to be receiving signals from the world), we will never know how "close" we are to "Reality".
3. One of the main problems of science is that we do not have a single clue about what the 'real' reality looks like, so that we can understand how close we are to the ultimate truth with our scientific theories. Withour being able to know what the goal is, it is very hard to know if you are going the correct way [1].
4. Science uses Logic as a tool to reach to conclusions. However even Aristotle, the founder of Logic, did not know how logic could be useful: as a tool to reach the truth or as merely a tool to analyze language and its structure? Many modern philosophers, like Wittgenstein, think that human language has many limitations and that due to these limitations, one must be careful as to talk only for things he/she can talk. The faith in the whole structure of science is based on the faith that logic works. If the latter collapses, then science is without any justification at all. One of the greatest mathematitians of all times, Russel, actually proved that logic has great limitations. The greatest logician after Aristotle, Godel, proved that science cannot prove it can prove things! Even logic requires faith to rely on after all...
4. Mathematics cannot spell numbers
From the beginning of human science, mathematics is considered to be the most crystal clear, provable, safe, well documented and better founded field of knowledge. Due to its nature, mathematics were the first field were an attempt to set the foundations of a fully justifiable / provable theory was conducted - and failed (see Hilbert program, Russel and Godel incompleteness theorem).
However mathematics have problems even in more fundamendal things. Try to ask a mathematician to write down the π number. He will start writting the first digits 3.14159... and then stop. You will ask him "why did you stop?" and he will answer "I stopped because π has infinite digits and we do not know all of them. We have also proved that its infinite number of digits does not follow any repetition pattern. So we cannot write all of them". But you are not satisfied...You want to know what π is.
The impressive thing about the above story, is that you will never get an answer and that mathematicians feel comfortable with that! Mathematics have named some numbers as "irrational" ("άρρητοι" in Greek), meaning that they cannot be expressed in writting since they have infinite digits that will never follow a pattern. One of these numbers is π. The square root of 2 is another example of irrational number.
Does that explanation is enough for the thinking human? No. When mathematics claim to be the most exact and well-founded scientific field, it sounds rather "basic stuff" to ask for a mathematitian to just write a number down on paper. But this is exactly what he/she cannot do! They say that we should be happy with the 1 billion digits we have calculated for π, but can we be happy when the "measurer" cannot measure?!?
This is more than just a good "trick" to play with. This shows an innate inability of mathematics to use its own language so as to just speak! And if we start with that, who knows what other inabilities we have overlooked?
5. Exact science does not understand reality
Exact sciences are a tool to formulate theories so as to explain what we see with our senses. Those scientific fields tells us nothing about what we call "reality". Reality is what it exists. We filter that reality via our eyes and ears and we then attempt to understand that "world of our senses" is.
SCIENCE RELIGION
Theory <=========> World of senses <=========> Reality
For example, things fall on the Earth. That is the reality. We sense that reality with our eyes and "see" apples fall onto the soil. We then try to explain what we see by formulating the theory of gravity. Many people think that since apples fall due to the theory of gravity and that since we see the apples, then the theory of gravity "exists". That is not correct: when the theory of gravity is proven wrong (that is happening all the time with all scientific theories), apples will continue to fall! Our theory, our interpretation of what we sense, will have changed, but that would have no effect whatsoever to the thing we call "reality". Another example is the invention of the transistor. The transistor works as the modern theories of physics say. However when all these theories are replaced by totally new ones, the transistor will continue to work...
The fact that reality continues to "work" has absolutely nothing to do with the fact that our models work or not. The former does not grant "validity" to the latter.
The fact that reality continues to "work" has absolutely nothing to do with the fact that our models work or not. The former does not grant "validity" to the latter.
IV. Failures of exact science
Science like physics and chemistry have failed to give explanations for many things we see in our everyday lives. Some of them include:
1. No scientific theory exists that requires 'causality' (i.e. that everything happening has a prior cause). As far as scientists are concerned, there could be things happening and their cause happening before them.
2. No scientific theory exists that requires 'time flowing forward' as we feel it happening. As far as scientists are concerned, the 'arrow of time' may as well be heading backwards. Some scientists today have attempted to explain that some scientific theories (like thermodynamics) really demand the arrow of time to go forward, but not with great success.
3. No scientific theory explains human goodnes, human altruism. The theory of evolution - no matter how good it explains many things about species evolution - cannot explain why you may endanger your own life to resque a complete stranger [see Evolution and Intelligent Design - The way to an agreement].
It must be noted that we do not examine here if science is correct or not in saying specific things. For example and as far as the point no.2 above is concerned, time could be one of our greatest illusions and science could be right in not finding evidence to support the "passing of time" (see Godel's circular and timeless universe on that). However this does not matter to my argument. What is important here is the innate inability of science to give a definite "yer"-or-"no" answer to questions, an inability too evident to be ingnored.
V. Success of exact science
Despite the above, fields of exact science are indeed the best tools we have to understand the physical world. Scientists today have created really good models of nature that predict and can help us live better lives, work more productively, understand Universe in much more detail than ever before.
VI. Conclusions - Going forward
We must use exact science carefully and always have in mind its limitations. Humans are more than electons and protons and this calls for the simultaneous use of other ways of thinkings - not only observation and induction. There are other ways that also help in searching for the truth. Believing in one thing only can be really dangerous. Believing that only the 'scientific' way os thinking exists can eliminate humanism, morality, altruism from the world. Many human values are not understandable by science. More philosphy and less 'scientistism' in todays society of gadgets will prove more than useful...
VII. Bibliography
1. "The meaning and limits of exact science" (Sinn und Grenzen der exakten Wissenschaft), Max Planck.
2. "Nature and the Greeks", Erwin Schroedinger.
3. "Philosophy - A graphic guide to the history of thinking", Dave Robinson and Judy Groves.
4. 'Tractatus Logico-Philosophicus', Lugwig Wittgenstein.
References
- 'Dissent Over Descent', Steve Fuller.
- Dictionary of Philosophy, Dagobert D. Runes, 1942, New York.
- Eliminative Materialism, Stanford Encyclopedia of Philosophy
- Physicalism, Stanford Encyclopedia of Philosophy
- Materialism's Slipping Hold on Science and Culture, Bruce Chapman, Seattle P-I, 1997
- Does Science Need Religion?, prof. Roger Trigg, Warwick University
- Mediocrity Principle [Wikipedia]
- Copernican Principle [Wikipedia]
- http://sthweb.bu.edu
/archives/index.php? option=com_awiki&vie w=mediawiki&article= User:Delta_x - Quotes of dogma from "The Observational Approach to Cosmology": (1) http://nedwww.ipac.c
altech.edu/level5/Se pt04/Hubble/Hubble3_ 2.html, (2) http://nedwww.ipac.c altech.edu/level5/Se pt04/Hubble/Hubble3_ 4.html, (3) http://nedwww.ipac.c altech.edu/level5/Se pt04/Hubble/Hubble3_ 6.html
The Observational Approach to Cosmology, Edwin Hubble, 1937, p. 50, 51 & 58.
Science New
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What is 'Religious Science'?
-James
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Science New
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Does 'exact' science represent either 'deterministic' or 'newtonian' science?
-James
And indeed modern science is based on determinism. I used the term "exact science" as I found it used by Max Planck in his work "Meaning and Limits of Exact Science" (Sinn und Grenzen der exakten Wissenschaft). I will try to find that book you mention and get back to you with more on that.
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Science New
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It is obvious that orientalism expnands a cope of science
Oriental culture is more adaptive to participate in religious science research than western culture, which relatively more relies on newtonian science. For example, as indicated in this article western modern science is mostly based on dualism initiated by Descartes while oriental philosophy such as 'Tao' is more natural oriented. One of examples for oriental culture based science research is new age science, which has been emerged from the late of 1960.
What do you think?
Do you mean that East-based science is beginning to play a major role?
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What do you think?
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Maria
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Thoughts / Reflections
I believe that morality, emotions and aesthetics can be measured, but not in a general way (i.e., all humans have the exact same experience of morality, emotions, etc.). For example, if chemistry does not provide us with a means for understanding emotions or moods, how can antidepressants possibly work? Is this a hoax? If so, then the pharmaceutical industry is bound to eventually colapse, right?
I think the only failure of science has to do with constraints or costs. Some things are too costly for people to measure exactly. It's to expensive to develop technology to measure things like tables exactly. However, when you look at the sciences in which people have a need for things to measured exactly (there is some economical benefit) usually humans progress towards better and better means for measurement until the return for such development tapers off (think of a growth function).
When you consider the economics of science, you then come to find that perhaps there are definite answers to questions, it just becomes a matter of whether humans will ever need these answers in order to continue to exist in our ever changing environment.
The understanding of chemistry reactions in the brain do provide information about exactly that: the chemistry of the brain. We cannot claim that we understand anything more than that. And I really cannot tell you what this chemistry means since we are not even close in understanding the real mechanisms of the brain. Perhaps that chemistry only deals with the results of our emotions or thoughts - so dealing with it is just dealing with the effects and not the real cause.
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However there is a basic difference between the "vaguety" of electricity voltage definition and that of anger or emotions. The difference is that our measurements of electricity can be applied and be used in everyday life, but no theory which entails measurement of emotions can be used in any specifically useful way in our lifes (to make predictions for example).
So finally you could say that there things "more measurable" than other", or that "some things are measurable as part of a prediction model, while others are not". Either way, the conslusion that our science - as it now exists - has limitations still applies.
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Your conclusion doesn't seem to be stated accurately. I would say that the correct conclusion from my comment would be "the scientific community may feel more confident in their measurement of some constructs than others." I think that further conclusions require the expertise of a researcher in the social sciences or some type of psychiatric studies. There are models for how people think that make use of some precise tools of measurement. Based on my experience I would say that the accuracy and scope of these tools has less to do with inherent measureability of the question, but the development of the constructs and technology used to measure them.
I think your conclusion: "our science - as it now exists - has limitations still applies." Is still too vague. If you differentiate between science (the theory and methods used) and the knowledge of the scientific community you will have an accurate conclusion. It is just linguistics.
For example, I may decide that perfection in running means a living creature can run 75 miles per hour. I live in the U.S. and I may never see this occur. It is possible for me to become frustrated and state that there are limitations to running in general because I have not seen anything living run 75 miles per hour. However, cheetas are reported to run 75 miles per hour in Africa. In this example, the physical construction of the cheeta meets conditions which allow it to run according to my ideals for perfect running. The living things in the U.S. have not. Just because some living creatures meet conditions which allow it to perform to my ideals and others do not does not mean that there are limitations to running in general. It simply means that there are principles which determine the performance of running.
Similarly, there may be principles which determine the performance of any scientific community. Some scientists may perform more according to ideals, some may not. This does not necessarily make any statement about the process of SCIENCE, rather it makes a statement about the conditions which affect performance in SCIENTIFIC ENDEAVORS.
I have worked hard to make my ideas clear in this post, I hope they are intriguing and clear.
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For me, I have pinpointed some major / essential characteristics of exact sciences to be:
1. Measurement.
2. Experiment.
3. Induction logic.
These are the things I refer to in the Knol and analyze their limitations.
Now I cannot tell whether you disagree with these specific limitations I mention, or if you think that there is no such distinction between "Exact sciences" and "Other sciences" (e.g. that psychology will some day manage to measure everything as physics does). Is that the case?
If yes, I must say that you will have a point when practice proves me wrong. Because until now no scientist in human sciences ("non-exact sciences") has managed to create resonable models that incorporate measurements as in physics or astronomy. Until now we have no idea how to mark an emotion as +2 or as -1,92. If we do, it will be because we ourselfs created a model that includes such numbers. But every model which refers to human thinking and/or emotions is bound to intefrere with the very essense of our existence: you cannot measure anger as a number because you cannot tell what is happening inside the human brain. You may now say that we may know in the future, but that is hardly an argument in favour of you: when we do know we can have another discussion. But now the fact is that these things cannot be measured.
Human thought and consiousness are things that seem to be subjective and very personal to each and every one of us. Modeling thought with mathematical models implies that if someone else reads these models will understand the thought of another person, as someone understands the "1+1 = 2". But this cannot be true. Even if I tell you "My anger is now +2", you will not have understood what I think and how angry I am. You will not have access to how my consiousness conceives that anger, how I REALLY feel inside. Again you may say that "someday we will have the tools to know". Then I would ask what data do you have so that you are so sure that it is a matter of current limitations of science and not of inherent limitations of science? And I would also assure you that when that day comes I will gladly change my view and admit that I was wrong. But right now the facts tell me otherwise.
I do not know if what I said was useful. Waiting for your feedback...
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OK, so one thing I notice you say is this:
"3. No scientific theory explains human goodnes, human altruism. The theory of evolution - no matter how good it explains many things about species evolution - cannot explain why you may endanger your own life to resque a complete stranger"
Well, I think that it is very hard for humans to intuitively come up with a reasonable theory because humans have many characteristics unlike other species. Humans are very successful in a range of environments and so to make a statement about humans in general is very difficult. We are diverse. However, I did find an interesting study done on a specific species of wasps: http://www.scienceda
I think the problem is that this question is very hard (expensive, technically complex) to test for humans, even if such studies were made longitudinal. There have been some theories and attempts at empirical testing: Hamilton's Rule has been tested (I think it was a Hamilton and Axelrod paper) and Game Theory presents the idea "Tit for tat." Maybe the explanation is that confidence in oneself serves specific function and that this confidence just happens to result in risking one's life for others. Afterall, isn't a man in a uniform attractive? Even if he risks his life for his nation... I will read your other post on Creationism in a bit.
Ok, I will also discuss human thought and consiousness a little bit. I think that it is possible for people to have very, very similar experiences and to be able to measure the similarity. This can be done by understanding much of the physical components to an emotional state. I read a bit on wikipedia about the physiology of anger:
http://en.wikipedia.
I think that two people with very similar physiological states would both be feeling similar feelings. (I would not call these feelings identical, just similar enough that "anger" or "sadness" was a word that was general enough to represent both people's feelings.) This is a hypothesis I would make based on the few studies I have read. I would need to have knowledge of how to use technology and knowledge of how to interpret the data derived from this technology in order to test this hypothesis with specific studies. However, I think studies of this nature can be done.
Hm, when I say: "I would not call these feelings identical, just similar enough that "anger" or "sadness" was a word that was general enough to represent both people's feelings." what I am talking about is the level of generality. (This appears to be a very important concept, though I have only heard of it once in a discussion group that I irregularly attended, so I cannot speak expertly, though I think I have a good intuition about it.) Using one expression of an addition alone is a very rough way to describe an idea. Using "1+1=2" is very, very general. It easily speaks to a large number of people. However, perhaps there is an expression in complex analysis that is, well, complex. This type of expression may speak to a large number of people AND it seems highly likely that this number is quite a bit smaller than the number of people who comprehend "1+1=2."
I think the same level of generality applies to conciousness. I am having a really hard time explaining my idea, so I will give you an example of how simple mathematics are useful in expressing complex emotion. The emotion I choose in this example is "wonder."
When I listen to Mozart ( http://www.youtube.c
Mozart expressed this emotion called "wonderful..." in a cryptic language known to the layperson as "sheet music." ^_^. In addition, there are codes and precise measurements that were used to support the operation of my computer, the internet, Youtube, etc. Many simple mathematical representations were used in the construction of all these man-made artifacts; and when combined in a very specific manner, result in an expression which inspires the feeling of wonder to millions.
So, the ways with which we describe human emotions can, in some sense, be described mathematically. There are differences in the how this emotion will be expressed. Some people will listen to it loudly,...
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Exactly how much can we reproduce? If we want, how similar can we make one experience as good as another? And will this reproduction be economically feasible? Will it be cost-effective? Because if it is, then it is just a matter of time before we can make better and better copies of things we want copied. And, if my theory that "people with very similar physiological states would both be feeling similar feelings" is accurate AND we humans are able to reproduce physical things with greater and greater accuracy, then it is likely that people will be capable of having increasingly similar experiences.
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Burçin Danacı
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Claes Johnson
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I agree: A coordinated choir is propagating the message that reductionist physics/mathematics is a model for all of science. But this song does not contain complexity and thus misses the essence
of scientific modeling of reality. It is not enough to stare at an elementary particle to understand the world. My main goal is to open to a synthesis of physics and soft science based on computation simulating the complex interaction of many agents.
Current physics seems to be a catastrophy from scientific point of view, completely removed from any rationale and reality. Maybe this is the prize for putting religion in the wardrobe?
Best regards,
Claes
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Anonymous
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Irrational numbers
Naming the things you don't know "infinity" does not mean that you actually "know" them...What is more interesting is that everything that can be done in mathematics with the use of the notion of infinity can be done without it also! "Infinity" is a man-made theoritical structure. And it actually contradicts reality in many ways.
Take for example the notion of a "straight line". Some mathematicians (not all) postulate that a line like that is constructed by an infinite number of points. But if we start to "cut" the line into an infinite number of small pieces, then we will end up with pieces that have zero length (that is the "limit" in our case). So is a line consisted of a sum of "nothings"?
In the same way, number Pi (π) is consisted of an infinite number of digits. However we don't have even a clue about where the "limit" of that infinite line of digits will be. We cannot say π is between 3.14 and 3.15, since that would mean that we simply ignore the fact that the next decimal digit signals us to search the limit between 3.140 and 3.142 and so on. If you want to fool yourself you can do that pretty easily. But if you are a real mathematician, you will try to find the place where that number will "settle in". And you will never can.
In that way, we may use number Pi as a monkey uses a computer to press a button and get a banana for reward. The monkey does not know anything about how the computer works. Neither do we know anything about Pi...
PS. You can easily construct a triangle with a hypotenuse equal to the square root of 2. However that number is irrational. In that way we are confronted with one of the major problems in philosophy: irrational numbers can and do exist in reality (e.g. on the piece of paper were we have drawn the triangle) but not in theory! Rather metaphysical I would say...
PS2. Expressing something you cannot understand in many different ways does not mean that you understand it. Expressing π as an infinite series of numbers for example, simply transposes the problem you have from "define Pi" to "tell me the limit of that series of numbers". Two ways to say the same thing: I cannot spell Pi.
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"What is more interesting is that everything that can be done in mathematics with the use of the notion of infinity can be done without it also!" This is a very bold statement that I find extremely hard to believe. Would you like to expand on this?
"In the same way, number Pi (π) is consisted of an infinite number of digits." Only if the expandsion of a number as a convergent series of powers of 10 is for some reason a definition of that number. Pi can be defined as the ratio between the circumference and the diameter of a circle. We can use this definition in many useful ways without ever needing to use decimal expansions.
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However Pi and all irrational numbers (like the square root of 2) have an interesting property: even though it is well defined in many different ways, it cannot be calculated as a value. It is not the definition I am arguing about, it is the calculation of Pi that I find as an indication of our inherent inability to grasp reality with the tool called "mathematics". You may define an infinite series of numbers and still be able to know that its sum is 5. You may define many mathematical constructs and then calculate them. But what if you define a construct and then you are not able to calculate it? What would that mean about our ability to understand reality? It is not a number that was imposed on us by some aliens, it is a number that as you said was constructed by us. And still, we can't tell where it stands on the line of numbers.
As far as infinity is concerned, it is true that using the notion of infinity you can reach conclusions that seem correct. And indeed Newton and Leibnitz used infinity (note: with the term "infinity" I refer to the infinately small terms 'dt' or 'dx' used in the Infinitesimal calculus). However even they were not so enthusiastic of that notion...
To start from the beginning, Democritus in ancient Greece talked about atoms and infinitely small "things" in space, time and geometry. However the paradoxes of Zenon made the use of infinity problematic and not all Greek mathematicians accepted that term. Euclid defined a point as "something with a position but with no size" in an attempt to exclude the notion of infinity from his works (infinity is not mentioned in his Elements). Aristotle excluded the term of "infinately small or big" from all geometry. Archimedes used the notion of "infinity" for his theorems in his work "The Quadrature of the Parabola", but then - since he did not accept the existence of such a term - he presented his results/proofs by using the method of exchaustion (that is, with the notion of "limit" and not the notion of "infinity"). That method is similar to the method of (ε, δ).
The one who insisted on using the term of "infinity" in mathematics was cardinal Nicolas de Cusa, because he thought that infinity was "the source, the medium and the ultimate purpose of every knowledge".
Leibnitz did not claim that infinity "existed" (as you say, it is a construct of human), only that it can lead us to correct results as if it existed...Neweton, even though he used infinity to reach his results, he then presented them in a completely Eucledian and finite way...
Bishop Berkeley (it is astounding how many people of the church use to do mathematics, despite many people believing that there is a "war" between science and religion) in his work "The Analyst" made an excellent criticism on the notion of "infinity" that was never refuted by mathematicians of his time. Moreover and in the 19th century, Weierstrass also proved that we can use the notion of limit without having to use "infinity" at all (something like the method of exchaustion of Archimedes).
Now, we ofcourse use both. Like Pi, infinity is still in the game but noone has ever concluded to whether that "thing" is related to something in what we call "reality" (see the Platonic Ideas World) or not. My point is just that we should not trust blindly science to lead us towards "reality", since not even mathematicians (the most clear-cut and theoritic scientists of all) have a clue on what the tools they use for thousands of years really "are"...
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John Gabriel
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What most professors of mathematics are completely oblivious to is the fact that irrational numbers have never been well-defined. You write, "...but can we be happy when the "measurer" cannot measure?!?"
The answer is: "of course not". Since Euclid, we have been approximating irrational numbers.
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Brian Fraley
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Jonatan Hägglund
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Thus: dealing with ideal values has not the same impact on mathematics as with , e.g., physics, since the mathematical realm is an ideal one...
One question that can be asked is: how can we even say that a number with infinite number of digits "exists" (even though we do not know what "exists" means, but that is another subject...), when the total number of particles in the Universe is about 10^70? How can we have a number of infinite number of digits when the whole universe does not have enough computational "capacity" to hold more than a certain number of digits if we used every particle as a bit of information?
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