23 February 2015

Thomson’s infinite lamp as a mathematical monster

Imagine that we have a lamp – it is switched off at its initial state and this state can be changed by pressing a button. Having been an hour to play with this lamp, we switch the lamp on after 30 minutes. Then after waiting for 15 minutes, we switch it off – and then we switch it on again exactly 7.5 minutes later – and so on. I think that the end of the story is self-evident: after one hour (and neglecting that it is physically impossible) we pressed the button infinitely many times.
But what will be the result? Will the lamp light? Or will not?
It seems to be an unanswerable question – after all, we can regard the switch off state as an “odd” and the switch on as an "even" number (or vice versa). The source of this problem is that only a natural number is either odd or even – but infinite is not a number in a traditional way.
But there is another analogy and it can help. Nobody knows PI’s exact value since it is an irrational number. What is more, according to our actual knowledge, its digits are randomly distributed. But if we would be able to compute all of its digits, would the last digit be an even number?
Perhaps it seems to be an acceptable answer that there is no a last digit of PI, so it is neither odd nor even. But PI is nothing more than the ratio of the circumference of a circle to its diameter and although we do not know exactly the numerical value of this ratio, it is a certain, existing value. Computing more and more digits of PI, we’ll know it more and more accurately – and computing it to the infinity, we’ll know it exactly.
Ad analogiam: if we press the button of the lamp infinitely many times, then the lamp will be necessarily either switched on or switched off – although we cannot predict the lamp’s state.
At this point we can distinguish to different types in math: random and compressible strings. The previous one means that we cannot find a representation of the given string which is shorter than the original one. Heads and tails is a good example for it: you won't know the result without tossing the coin in reality.
Or onecould mention the cellular automatons (CAs). The state of their cells depend on the neighboring cells’ states and a CA changes in discrete steps. The result is that although the system is absolute deterministic (certain starting configurations always results the same next phases), cellular automation is an incompressible process. We cannot compute the next phase without executing the program itself.
Opposite to these above mentioned examples, a compressible string can be regarded to be “regular” in a sense that if we know the rule, then we can find the nth digit without computing others.
Our lamp represents a totally different solution. We can compute its every stage and its algorithm is ridiculously simple, so it is compressible - except for its endpoint. We cannot answer whether the lamp is switched on at its final stage – unless we de facto pressed that button for infinitely many times.
I wonder whether there are other, strange categories – for example, who could imagine a string which is compressible only at its endpoint? Perhaps other mathematical monsters lurking somewhere.

17 February 2015

How (not) to avoid the G-world in cosmology?

The first chapter’s title in Mary – Jane Rubinstein’s recently published book entitled Worlds Without End. The Many Lives of the Multiverse is: How to Avoid the G-word.
Namely, the God.
It is an interesting trend that not trying to avoid the G-word is an acceptable attitude for some modern cosmologist, although the Creator’s existence or non-existence hasn't play a role in modern optics or mechanics (what is more, it is not a question in the explanation
At the beginning of the 17th century Galileo argued that there were two books written by God. One of them was the the Bible and another was the book of nature. Although it was a common belief of that age that both of them used the same language, according to Galileo, the second one used the language of mathematics. [John Hedley Brooke: Science and Religion, p. 104] Supposing the existence of a Creator who created the World for humans, it is an open question why He had decided to use two different languages, and why an exotic and complicated system was introduced to describe the physical reality created by Him. In other words: why the physical reality is so complicated that it is impossible to describe it in everyday words? Was it a necessity to hide the structure of the world behind a complicated and non-evident formulism?
Obviously, Galileo’s concept based on a kind of Neoplatonic mathematical mysticism. According to it, the fundamental structure of our Universe is not only can be described by mathematics, but the nature itself is purely mathematical. But it is not an answer for the question why decided God to choose this language.
On the other hand, a group of physicists in the 17th century refused the Aristotelian concept which stated that an explanation cannot be complete without pointing out a final cause. Since Aristotle didn’t favor mathematics as a language of world description, Galileo’s approach offered a way to deny any argumentation based on final cause, and Descartes decided to focus on immediate causes of physical events. [Brooke, ibid, p. 72]
Since then it became a tradition in natural sciences to study those immediate causes and omitting either a final cause or the intervention of a Creator. But – to oversimplify the problem – when somebody studies either the “fine-tuned” nature of our Universe or the supposed existence of multiverses, there isn't an opportunity to study his or her subject from Descartes’ approach, because there is no an existing physical event to examine.
According to Bernard Carr, “If you don’t want God, you would better have a multiverse.” Namely: if you don’t want to refer to God, then you can replace Him with the idea of Multiverse – or vice versa. The “Multiverse replaces God with what is perhaps an equally baffling article of faith: the actual existence of an infinite number of worlds.” [Rubinstein, p. 29]
We have two different strategies to handle the causes. One can choose Descartes’ approach to always focus on immediate ones – and then he or she can ask another question about the solved problem’s immediate cause. We can move forward step by step – its mathematical analogy is the potential infinity where every piece of the result is finite and it can be achieved in a finite number of steps, but this process never has an endpoint.
The other approach is more problematic. If we are allowed to ask for a not immediate cause, then we could ask what caused that cause, etc. [Rubinstein, p. 21] and the result is an infinite regress without end.

13 February 2015

Readings: applied philosophy of science II.

 Part I. of my short reading list contained some introductory books about philosophy and philosophy of science. Let’s see now some readings about the philosophy of mathematics; and cosmological and other works worth to mention. I am to underline again that this list is partly about my personal tastes, but undoubtedly useful for those are interested in this field without any previous experience. The only skill needed to enjoy these books is some affinity and tenacity.


1. Körner, Stephan: The Philosophy of Mathematics. An Introductory Essay (Dover Publications 2009).
This compact and comprehensive volume was published originally in 1960, but it is not obsolete even today. Sketches about Plato’s, Aristotle’s, Leibniz’s, Kant’s views; foundations of pure and applied mathematics; expository and critical chapters about the main modern schools of mathematics (formalists, logicists, intuitionists). 

2. Hersh, Reuben: What is mathematics, really? (Oxford Univ. Press 1997)
More detailed than Körner with chapters about the “Criteria for a Philosophy of Mathematics”; the role of intuition and proof; misleading mathematical myths (unity, universality, certainty, objectivity of mathematics); “Humanists” and “Mavericks” in mathematics etc.

3. Barrow, John D.: Pi in the Sky. Counting, Thinking and Being (Little, Brown and Company 1992)
Interpreting the subject in a broader sense and mixed with history, general philosophy, even theology. Connections with other sciences (i.e. computability and compressibility); mathematical heroes to Gödel and “transhumanist” and “Marxist” mathematics… This book is not only about the philosophy of mathematics, but about how it is embedded into our culture and natural sciences.

4. Wolfram, Stephen: A New Kind of Science (Wolfram Media 2002)
online: http://www.wolframscience.com/nksonline/toc.html
One of the most exotic and controversial book about the rethinking the foundations of mathematics from a computational point of view. Wolfram uses “cellular automatas” (CA – it is a deterministic dynamical system) to model different processes offering an alternatives for the traditional mathematical/modelling approach. Although some of his generalizations are questionable in connection with biology or with the dynamics of society, the conclusion that “simple programs can produce great complexity” offers a new interpretation of both mathematics and natural sciences.


5. Ellis, George F. R.: Issues in the Philosophy of Cosmology (2008)
online: http://arxiv.org/pdf/astro-ph/0602280v2.pdf
A short and deep survey about the main themes of philosophy of cosmology from the uniqueness of universe to the question of its origins to multiverses and the nature of existence. If one reads this paper (which appendix contains a summary table about the main thesis and issues), then he/she meets every fundamental problem of cosmology.

6. Carr, Bernard (editor): Universe or Multiverse? (Cambridge Univ. Press 2007)
Reading into some frontiers in cosmology – this volume explains this subject from both philosophical and physical approach. Since the multiverse proposal is one of the most exciting questions of this field, almost every subject from Anthropic Principle to the concept of a mathematical universe based on Platonist ideas appears in this volume. The different chapters are written by different authors, so sometimes you find equations, but these studies are both understandable and readable.

7. Lem, Stanislaw: Summa technologiae (Univ. of Minnesota Press 2014)
One of most unique book about the possible future of science and technology. Written more than a half a century ago by Polish science fiction writer and thinker Stanislaw Lem, it begins with an interpretation of evolution as a kind of universal problem solving process, and continues with diverse and exotic subject from cosmic civilizations to “intelectonics” (intelligence+electronics); phantomology (virtual reality), etc. A critics of this book compared Summa technolgiae to Thomas Aquinas’ Summa theologica, and it gives a superb example for creative thinking inspired by the promises of technology and science.

8. Stapledon, Olaf: Star Maker (Dover Publications 2008)
Another SF author of this list – with a science-fiction book. Stapledon writes about the life in universe and its strange, almost unimaginable form and ideas and technological solutions which inspired the next generation of thinkers form Arthur C. Clarke to Dandridge Cole to Freeman Dyson. Beyond the fact that it is amusing science fiction work with philosophical insights, it is a gold mine for those hunt for new ideas.

09 February 2015

Readings: applied philosophy of science I.

So do you want to study the problems related to this field?
Basically there are two kind of reading lists. The first one is for those who want to examine a certain field – i.e. the applied philosophy of science (after all, what else:-). The second one is for those whose aim is not to become an expert of that field, but intend to apply its results. This list belongs to the second category: I am aware that reading Kuhn or Popper (not to mention Merton or Hull or Aristotle) is a really enjoyable form of either recreation or scientific research. But it is not necessary to apply their results. So you do not need to read all the classics – it is enough to understand them to an appropriate level.
I collected some readable and easy to understand, but serious titles. They are usually available either in bigger libraries or via an online bookshop. Their subjects are divided into about five groups: general philosophy; philosophy of science; philosophy of mathematics (as the traditional natural sciences based on it); SETI; cosmology and miscellaneous readings.
My approach is mainly physics centered, but I hope that it is a useful list even for those who are interested in, for example, evolution. Obviously, there are other, equally good works about these topics, but they are my favorites, so this list at least partially about my personal preferences. My advice is to read them in the following order:


1. Baggini, Julian – Fosl, Peter S.: The Philosopher’s Toolkit. A Compendium of Philosophical Concepts and Methods (Wiley – Backwell 2010)
In accordance with its title, this volume is about philosophical tools and their usage from axioms and hypothetico-deductive method to the meaning of a priori; self-defeating arguments and Hume’s fork. Use it as a thinking toolbox – very efficient.

2. Warburton, Nigel: Philosophy. The Basics (Routledge 1999)
Survey about “general beliefs” from the meaning of the life to the existence of God and the design argument; common-sense realism; negative freedom… etc. This book discuss these basic themes from a non-historical, problem oriented point of view.

3. Scruton, Roger: Modern Philosophy. An Introduction and Survey (Mandarin 1994)
It discusses more subjects than typical for introductory texts from intentionality to modality and space and time in a lucid and readable style. Offers a deep and sympathetic understanding.


4. Okasha, Samir: Philosophy of Science. A Very Short Introduction (Oxford Univ. Press 2002)
What is science? Problems of reasoning, explanation, realism and anti-realism and philosophical problems of physics, biology, psychology... Short and compact summary of the main problems of philosophy of science – a good starting point to understand some basic concepts.

5. Godfrey-Smith, Peter: Theory and Reality. An Introduction to the Philosophy of Science (Univ. of Chicago Press, 2003)
Deep and detailed guide to the 20th century’s developments from logical positivism to Feyerabend, Latour and feminist criticism. It was written for students learning philosophy of science, but accessible to a reader with general interest without any background in philosophy, as well.

6. DeWitt, Richard: Worldviews. An Introduction to the History and Philosophy of Science (Wiley – Blackwell 2010)
In accordance with its title, it is about both history and philosophy of science. The story begins with Aristotle and his “grocery list” of beliefs, and ends with quantum theory and locality. DeWitt shows why a certain worldview (either Aristotle’s or Tycho’s) seemed to be acceptable to their contemporary thinkers, and offers an inner picture about these approaches. History and philosophy is interwoven in it, and it is not an accident, since both of them are required to understand the process of science.

7. Henry, John: A Short History of Sceintific Thought (Palgrave – Macmillan 2012)
History of scientific thought as a fuel of philosophy of science. The emphasis is on the developments of science; a readable style with a lot of piece of information – even a historian of scientist would find both new data and contexts in it (unless he/she is an expert of the whole history of science from ancient Greek philosophers to Hutton and Baresh Hoffmann – but it is not too probable).

to be continued...

05 February 2015

Popper’s bridge - and why it repeats the induction fault

To oversimplify it: according to Popper, induction is not an acceptable scientific method, since a finite number of observations is unable to verify the truth of a statement. It is always possible that we would find a counterexample in the future. So the solution is falsification, and passing more and more observational tests is not a cause to increase our confidence in a hypothesis.
Now imagine that we have to build a bridge and we have two physical theories: an older one which is “tested many times and has passed every test… and a brand new theory that… has never been tested." Applying Popper’s logic, there is no an essential difference between them: neither of them falsified, so they are equally “reliable.” 
 Obviously, it would be simply stupidity to choose the newer hypothesis, although Popper didn’t have a good answer for this contradiction [Godfrey-Smith, Peter: Theory and Reality 2003, p. 67 - 68].
Popper introduced his concept to bypass the problems of induction. The bridge problem pointed out the weakness of a theory based solely on falsification. It is a similarly serious counter argument that Popper’s falsification concept based on induction, although his aim was to eliminate the logical problems of it.
He stated that a counterexample can disprove a theory. In other worlds: the result of a sole observation can be extended to every future incidence. But this presumption based on an induction, namely that if we repeat the observation than it will necessarily disprove our theory again. But following Popper’s logic which refuses the validity of induction, falsification can prove only that a model/concept/etc. actually don’t work, but cannot prove that it won’t work in the future – after all, Popper’s central idea was the rejection of induction.

03 February 2015

Creating universe hierarchies

There are different and at least theoretically possible ways of universe construction– and we can repeat them.
The most trivial solution is creation by natural laws – it is presumably a relative slow process from a declared beginning (in the case of our Universe: from the Big Bang) to the appearance of stars, galaxies and intelligent life. It is another question why needed a so huge time for it.
Another possibility is a creation by creator – either in the form of computer simulation or by creating a universe using the laws of physics.
A third solution is a genesis thanks to vacuum fluctuations – in this case the Universe can appear instantly both with its whole complexity and with its thinking inhabitants.
It is imaginable that not only a hierarchy of universes exists where a whole world is created within another one, but the origins of the universes on different levels are either the same or different. A naturally evolved universe can be a home of an intelligence which creates an artificial world (say, by natural laws) which existing for an infinitely long time results the rise of a new universe with observers by quantum fluctuations, etc.
So our first (and trivial) conclusion is that is that regarding the “tools” which was used (natural laws, chance, etc.) we can distinguish natural and artificial origins – including into the latter category both a God-like creation and a creation by programming. A creator of a programmed universe is restricted only by the laws of logic: for example, he/she aren’t able to create a circled square. The opportunities of a creator of a physically existing universe is more problematic – obviously, his options determined by the natural laws. The main question from this point of view is whether a “world creation” means “only” the creation of the world as an object – similarly to a sculptor who formats the clay? Or, his/her activities includes the forming of the natural laws which rules the new universe, as well?
The second, similarly trivial conclusion is that there are two basic type of the hierarchies of universes: a homogenous and a non-homogenous. A homogenous universe hierarchy can be described as a chain of embedded worlds where the first one was created by natural processes; and then thanks to natural processes, too, a new universe appears within the first one, etc. Lee Smolin’s “cosmological natural selection” is typical example for it.
Obviously, mixed solutions are imaginable, as well. A universe with natural origin containing naturally evolved intelligence which creates worlds; an artificial universe where new universes arises because of natural processes, etc. Realize that this distinction between “natural” and “artificial” worlds are merely historical, since an outsider observing the creation of the hierarchy of universes can specify them, but it is undecidable if you live in it.