Up-to-date Mental Representation Methods
Version date: July 4, 2008
This easy-to-read text proposes an update of mental representation methods of reality that enables persons without scientific background to go beyond the simplistic concepts of traditional determinism, the determinism of Laplace.
This text is a conclusion to my book "Determinism Extended to Better Understand and Anticipate – A Bridge between Science and Philosophy for Rational Thinking". It provides details on the notion of randomness, and an answer to the question "What can we understand using rational thinking?"
For details and justifications of this text's statements, see the book above.
Many people divide nature's phenomena into two categories: those that may be explained by laws of nature (the phenomena called deterministic in the traditional sense), and those due to chance (the random phenomena). They believe that the latter cannot be explained, that nature is unpredictable in their case, and that two identical situations may evolve differently.
I believe that those people describe too often as random a phenomenon that they misunderstand; instead of admitting their ignorance, they describe nature's ways as unstable. This text provides precisions about the limits between determinism and chance, and offers models to represent nature that take into account modern science.
Replacing random positioning by an area of fuzzy presence
The book describes many examples of the erroneous nature of traditional determinism. It also describes cases where the description of nature is necessarily probabilistic and probabilities are indispensable for every representation of reality. This is the case, for example, of Louis de Broglie's matter waves, which associate a probability wave with every moving object: a moving object is accompanied by a packet of probability waves, and its position at a given time cannot be known except using a function that defines the probability density of presence at each point of space.
The probability density (probability by unit of volume) d at point M is a positive number with which one can calculate the probability of presence p(M, ΔV) in a volume ΔV around M using the formula p(M, ΔV) = d . ΔV.
Matter waves can penetrate matter or potential energy barriers much as sound waves penetrate solid walls: inside such a barrier, the probability of presence of a moving body is non-null. The barrier crossing or penetration phenomenon is very frequent; it is termed tunnel effect.
The experimental proof of the existence of matter waves was observed in 1922-1923, when the Compton effect was discovered. As a consequence of this effect, the position or the dimension of a moving object with mass m cannot be defined with an accuracy better than h/mc, a quantity known as the Compton wavelength lc (where h is a constant of the Universe known as Planck's constant, and c is another constant of the Universe, the speed of light in a vacuum). The width of the packet of matter waves that accompany a moving particle is approximately equal to lc, so it is illusory to attempt to define its position or dimension with an error less than lc. The existence of lc compelled physicists to abandon another certainty that seemed intuitively obvious, the certainty that an object's position or dimension may be arbitrarily precise: there is a minimum uncertainty.
We should not interpret a moving object's probability wave at time t as a necessary ignorance of its position due to chance, but as the fact that a precise position is an oversimplifying abstraction of the human mind, an abstraction that misrepresents reality at atomic scale. The concept of precise position at time t is replaced with an area of fuzzy presence, where each point of space has a probability density of presence.
Nature appears unpredictable only if we demand that it behaves with a precision that it cannot provide. Instead of considering its position at time t as random, we should consider it as fuzzy, in the vicinity of a point of maximum probability of presence.
In short, instead of talking about randomness, we should alter our representation of reality; for a moving object, we should replace the notion of precise position with a notion of presence area whose limit is fuzzy, an area where the probability density of presence, maximum at a certain point, decreases rapidly with our distance from it.
Imprecise dimensions and fuzzy limits
The fuzzy position of a moving object is not the only imprecision at atomic scale, where some contours are also imprecise, therefore having imprecise dimensions.
§ A proton's radius is imprecise, about 0.8 10‑15 m. It is a sort of ball of positive electric charge, surrounded by a "skin" where this electric charge decreases very fast when the distance to the center increases.
§ The size of the nucleus of some heavy atoms is about 10‑14 m. This size is imprecise because its shape keeps changing: such a nucleus is constantly deformed by the agitation of its protons and neutrons. This agitation may become so intense that it exceeds the resistance limit of the nucleus, which then spontaneously breaks, producing smaller nucleuses. This happens, for example, when an atom of uranium 238 decays.
We should therefore imagine the surface of some atomic particles like the surface of a star, a gaseous mass with a fuzzy external limit constantly deformed by coronal mass ejections. Huge stars and tiny atomic particles both have blurred contours and imprecise dimensions.
Imprecise velocities and coupled uncertainties
In addition to the inaccuracies on a particle's shape and position, there is an inaccuracy on its velocity, a variable also defined by a wave packet. And last but not least, quantum mechanics shows that a simultaneous measure of both variables of certain couples, such as position and velocity or energy and time, has a minimum uncertainty: the product of the uncertainties in position and velocity, or in energy and time cannot be less than h/4p; so if the inaccuracy on a particle's position is small, the inaccuracy on its velocity is large, and vice-versa. This limitation is known as Heisenberg's uncertainty principle, and it shows nature's refusal to abide by our preconceived ideas about the precision of measurable variables. Whether we like it or not, nature's precision is limited when it concerns variables of motion.
Allow for discontinuity in addition to continuity
Physicists tried unsuccessfully for many years to find a law describing the emission of heat by a blackbody as a function of its temperature. They were trying to calculate the quantity of heat emitted as electromagnetic energy such as infrared light. As long as they tried to fit a continuous mathematical law to the physical phenomenon, they failed. In 1900, Max Planck showed that the quantity of heat exchanged by waves of frequency n is always a multiple of a minimum, hn, where h is Planck's constant.
Planck's discovery implied representing the quantity of energy exchanged at frequency n by a discrete value, a multiple of hn, instead of a continuous value that would have allowed each exchange to be arbitrarily small.
Physicists had to abandon their usual continuous functions, so easy to understand intuitively and manipulate mathematically, in favor of a non-intuitive representation; the approach required the same openness of mind as replacing the concept of precise position with a concept of fuzzy presence area with blurred contour. A better understanding of reality was obtained at the cost of a slightly more mathematical representation of its phenomena.
Complete Newton's second law of motion with the Schrödinger equation
In classical mechanics, a force represented by vector F, acting on a body with mass m, imposes to it an acceleration represented by vector a such that F = ma. This is the best-known equation of Newton's physics, and it is a differential equation since the acceleration is the second order time derivative of the body's position. It is a consequence of the conservation of total energy (kinetic energy + potential energy) when a body moves in a field of force.
The same physical law of total energy conservation is valid at both atomic and macroscopic scales. When combined with Louis de Broglie's matter waves, better suited to describe the position and motion of a tiny particle, it predicts a trajectory described by the Schrödinger equation, the differential equation that replaces Newton's at microscopic scale.
The Schrödinger equation uses more advanced mathematical tools than Newton's equation does: Hilbert vector spaces, operators and their eigenvalues. Its solutions have the same probabilistic nature as the matter waves it derives from; the notion of trajectory-sequence-of-points of Newtonian mechanics is replaced with a moving wave packet: at any time t, the position of the moving particle is defined by the fuzzy contours of that wave packet.
Energy fluctuations, non-separability, tunnel effect, etc.
The principle of energy conservation seems so obvious that it is accepted intuitively. However, energy may fluctuate at each point of a vacuum, "borrowing" from the surrounding empty space enough energy to create ephemeral particle-antiparticle pairs, whose annihilation returns the energy to the vacuum after a very short period of time. This phenomenon, frequent at atomic scale, also occurred in the primitive Universe, where it produced the energy density variations that later gave birth to galaxies; it continues to occur today when black holes "evaporate". We must accept it in spite of its counterintuitive nature, that leads us to believe that "it is impossible to create something from nothing" or, equivalently, accept that vacuum may contain energy.
We must also accept that some physical group properties of a pair of particles born together (such as two photons with opposite polarizations) may remain unchanged even if these particles become separated by several kilometers. In other words, the notion of spatial separation of two particles does not apply to all of their properties; any action that alters one is immediately reflected by an alteration of the other, in zero seconds, regardless of their distance. This property of our Universe, termed non-separability, is inconceivable in traditional physics; sometimes determinism applies to a group and not to each of its members considered separately, and its consequences are then propagated instantaneously, not limited by the speed of light.
We must also accept the tunnel effect, a probabilistic consequence of matter waves which enables a particle to travel through matter or through a barrier of potential for which its energy will not suffice. We must accept it because it is so real that it is used in some transistors.
We must also accept that many molecular bonds of biochemistry have a probability of appearing and a probability of breaking apart. Such probabilities, accounted for by quantum mechanics, cause genomic replication "accidents" responsible for mutations of species; they also explain that some populations have genes that adapt their digestion mechanisms to food available locally; they also explain the occasional presence on a single gene of one extra methyl (CH3) radical that inhibits the gene's expression.
There are many physical phenomena that challenge our intuition, and that we must get accustomed to accepting without attributing them to chance, to a supernatural cause or to nature's unpredictability. Scientific reality is often more surprising than fiction.
The effort required to adapt our mental representations
For a man with modest scientific background, the effort required to adapt his mental representations to the results of quantum mechanics is similar to the effort required to accept the results of Newtonian mechanics. He only needs to know:
§ That some physical properties such as the shape, position or speed of a moving particle are defined by nature with some fuzziness, just like the blurred image seen through binoculars that are out of focus, where objects' contours are imprecise;
§ That some measurable quantities, continuous at macroscopic scale, are discontinuous (quantized) at atomic scale. An example is a particle's energy.
Stop believing in randomness as a principle of unpredictability or instability
From a philosophical point of view, the first priority is to stop believing in chance as a principle of unpredictable behavior of nature, or of its instability. The Schrödinger equation is perfectly deterministic in the traditional sense, just like Newton's law of motion which has the same basis, energy conservation: the same initial state always produces the same final result; no unpredictability there.
At time t there is no randomness in the position or the velocity of a particle; there is indetermination, which is nature's way of refusing to grant us the possibility of infinite precision which would satisfy our mind, a refusal due to the undulatory nature of each particle.
Finally, the many examples of chaotic phenomena described in the book confirm the limited precision often permitted by nature.
Conclusions:
§ Let us not confuse chance (randomness, unpredictability) with indetermination (nature's refusal to be precise);
§ More generally, as explained in the book, determinism and predictability are different concepts: the latter does not necessarily result from the former.
Accept that a cause may produce several consequences at atomic scale…
At atomic scale, nature replaces the single evolution solution of Newtonian physics, by a set of evolution solutions, the points of a fuzzy region of space where each point has a probability of presence at each time. This results from a mathematical property of the fundamental model of quantum mechanics, the Schrödinger equation, model whose accuracy of representation of reality is perfect and proven by countless experiments.
The solutions of the Schrödinger equation happen to be probabilistic matter waves. The randomness introduced by nature at that scale is similar to that of a dice throw: the result is deterministic, it is always the same set of solutions (a number between 1 and 6), from which nature finally chooses a single element that may not be known before the dice stops rolling.
Therefore, the second effort required from us is accepting that a unique well-defined cause (such as throwing a dice) may produce a consequence which is a set of values instead of a single value. The consequence-set is still always the same for a given cause, but it is a set that may have several elements, often even an infinite number.
After producing the consequence-set of a given cause, nature chooses one element when measuring a variable, and that is where randomness intervenes: nature will not let us know which of the 6 values will appear before it does. Nature chooses one of the solutions of its consequence-set during the transition from atomic scale to macroscopic scale. This transition is an extremely brutal process: a measuring device used in quantum physics amplifies energy billions of times to make it perceivable by a human; and this transition is an irreversible process. The third effort required from us is to understand that randomness intervenes only in the choice of a single solution, not in its measurable value, and that it may be due to the brutality of the change of scale during the measure.
…And also at macroscopic scale
Multiple consequences of a single cause also occur for macroscopic phenomena. This is the case, for example, for turbulent flows in fluid mechanics. It is also the case for thermodynamically unstable systems that dissipate energy, such as living beings; such systems evolve towards points of their phase space termed "strange attractors", a phenomenon that explains Darwinian evolution of living species towards increasing complexity. Randomness intervenes only when nature chooses one of the consequences among all those that are possible; this is a choice of a value belonging to a predefined set, not a choice of the numerical value itself.
Conclusion on the plurality of consequences and precision
We should stop expecting from nature a unique consequence of each cause, a behavior it refuses in many cases; such a behavior has no relationship with fantasy or man's ignorance: nature sometimes refuses a unique consequence (which would require a unique solution of an evolution equation), exactly as it refuses to display a precise contour for tiny particles or giant stars. For some phenomena, we should also renounce precise results of measures or predictions.
Nature is not obligated to adapt to the simple mental representations we prefer; it is up to us to adapt our representations to nature's reality.
Determinism governs the evolution resulting from the initial cause; it does not guarantee that the result of this evolution may be predicted before or measured afterward with arbitrary precision. Causality is always respected, but it does not guarantee the predictability of the result or the precision of its measure.
Accept the consequences of Relativity at astronomical scale
At astronomical scale, we should get accustomed to think of space and time as interdependent within a four-dimensional space-time continuum where each point represents an event. We should also admit that space itself is locally deformed by the presence of heavy masses such as a galaxy, a black hole or a star; this deformation makes light follow a shortest path which is curved.
We should also realize that the order of precedence of two events A and B is not always the same, since A may precede B for some observers and B may precede A for others. Finally, we should accept the fact that the Universe expands, the galaxy clusters moving apart progressively from one another, and that 96% of its energy takes two forms that are invisible and as yet unexplained; the first is termed "dark matter", and the second (which creates a negative attraction force) "dark energy".
The Relativity theory that governs all those phenomena is a representation of reality confirmed by many experiments, and we can no longer ignore its implications.
Adopt hierarchical representation models for complex realities
The human mind cannot manipulate more than half a dozen concepts at the same time. To comprehend a complex phenomenon, our mind needs to represent it using a multilevel hierarchy. Each level schematizes the concepts of the level below it, thus making its structural and functional organizations simple enough to understand.
Building a representation of a complex phenomenon may use a bottom-up approach, with a succession of syntheses and schematizations that hide the details to highlight the essential concepts. Or it may use a holistic approach, studying one of the levels – or one of the phenomena of a level – as a whole, with the same goal of reducing its complexity to an acceptable level.
(A top-down approach – from general features to their details - is not suitable for studying and understanding a complex phenomenon. It is suited for describing and explaining it to someone else when it is already understood by the person who explains it.)
Use schemas to structure knowledge
Using schemas helps our mind understand a concept such as an imprecise dimension or a fuzzy contour by associating it with an image such as a sphere of mist. And our mind can picture the probability distribution in the vicinity of a point as a bell-shaped curve.
One of the best ways to represent a complex object is using an analogy with the structure of a complex piece of software. The architecture of the software's modules simulates the logical relationship structure of the object's physical structure, and the behavior of the software's modules simulates the object's functions. This model is the best for understanding man using a hierarchy of software modules from the genome level to the psyche levels.
Use tree hierarchies to represent possible causal structures
Another reason for adopting a hierarchical representation of natural phenomena is determinism itself. In addition to local determinism, which governs the time relationships between causes and consequences at individual cause level, there is a global determinism which governs global phenomena such as: the choice of an entire trajectory of a material object that moves between two points; the choice of the path of a light ray across several media; or the choice of the macroscopic thermodynamic evolution of a system that comprises billions of molecules. Global determinism never contradicts local determinism; it complements it, providing an elegant way of understanding a phenomenon's "big picture".
Sample local determinism: the principle that governs the step-by-step choice of a moving body's next small displacement, following the differential equation of Newtonian dynamics F = ma. At global level, the same phenomenon follows Maupertuis' principle of least action, which "chooses the best trajectory" (the path with minimum action) between a start point and an arrival point.
The possibility of multiple consequences for a given cause requires replacing the unique causal chain of Laplacian determinism with a hierarchical causal tree. At each stage of a system's evolution, represented by a node of the tree, nature "chooses" the next evolution process, represented by a branch starting from this node. The global representation of possible deterministic evolutions from an initial situation is a tree.
Overlooking levels hinders understanding
Some philosophers reject the materialist explanation of the world because they find explaining concepts as rich as human personality impossible starting from molecular structures and functions. Indeed, such an explanation is impossible in a single stage, because a multilevel abstraction structure is indispensable. The issue is the same as explaining the architecture of a complex software application, which requires successive levels of detail descending from the highest level, oriented towards the user, to the lowest, which is also the most technical and suitable for execution by the computer.
In addition, in the field of living beings, acquiring and explaining knowledge requires from time to time a holistic approach that considers a specific object or function as a whole whose interaction with its environment is simple and well delimited.
It is therefore obvious that descriptions of psychic phenomena, or even biological phenomena, based directly on physical properties are impossible; but that does not prove that materialism should be rejected: a proof of the absurdity of a doctrine based on an erroneous approach does not prove anything about the doctrine.
The limits of comprehension due to methods
Here is a concise description of methods used to study phenomena by building a representation to understand them, and then anticipate their outcome; such methods have intrinsic limitations.
§ Mathematical modeling of physical phenomena
Example: the measure of a physical quantity results in multiple solutions associated with the various eigenvalues of the operator that represents the quantity; and each eigenvalue has a probability if it is discrete and a probability density if it is continuous. This approach results in multiple potential consequences of an initial cause.
Modeling an evolution with the Schrödinger equation provides perfect results: it is impossible to discover another model that would produce fewer solutions or non-probabilistic solutions, since nature itself conforms to this model.
§ Uncertainties such as those due to Heisenberg's principle
Heisenberg's uncertainty principle is in fact a theorem which is rather easy to prove. It is a consequence of a mathematical property of certain couples of operators that represent physical variables, operators that do not commute. It states the physical impossibility to determine the values of both variables simultaneously with an arbitrary precision level: the more precisely one of the variables is measured, the less precisely the other is known.
An important consequence of that uncertainty is energy instability (fluctuations) at atomic level. That instability should not be confused with thermal restlessness (random movements due to absolute temperature being greater than zero).
§ Wave-corpuscle duality
Depending on the experiment, the same particle may behave either as a corpuscle or as a wave.
· A photon is an electromagnetic wave of short duration whose behavior may cause interferences. It is also a particle without mass, whose shock with another particle such as an electron or an atom may transfer momentum and angular momentum – sometimes with enough force to displace it or even to shatter it.
· An atom is a corpuscle with non-null mass. Louis de Broglie's theory of matter waves shows that it may also behave like a wave. Even though that wave is a probability wave, not an electromagnetic wave, it may also produce interference patterns with the matter wave of another corpuscle. And the shock of a corpuscle with another or with a photon may transfer momentum and angular momentum.
Depending on circumstances or experiments, the perception of physical reality may change.
§ Undecidability, incompleteness and inconsistency of an axiomatic system
It is often worthwhile representing a scientific field using an axiomatic system, to subsequently study it by deducing logically some of its properties as predicates. But this approach has limits: no matter how an axiomatic system is defined, it will have two serious limitations, undecidability and uncertainties about its consistency (non-contradiction).
· A logical proposition is a statement which is always true or always false. Every axiomatic system allows enunciating some logical propositions for which no proof of their truth or falsity may exist: it is impossible to prove that one such proposition is true or false using the axiomatic system's axioms and deduction rules.
Such a proposition is termed undecidable. There are many undecidable thoughts in a human mind that it considers certain, and their presence cannot be explained in a deterministic manner: it cannot be attributed to any effective cause or stable set of circumstances. Fortunately, if factual observations show that a given proposition of an axiomatic system is true and never disproved, the system may be completed with this proposition, admitted as a complementary axiom.
· An axiomatic system is termed complete if every logical proposition that may be deduced from its axioms and deduction rules is decidable. But since every axiomatic system allows enunciating undecidable propositions, no axiomatic system is complete.
· An axiomatic system is termed consistent if every theorem deduced from its axioms is itself non-contradictory, and does not contradict any other theorem or axiom of the system. Kurt Gödel proved the impossibility to demonstrate the consistency of an axiomatic system as a theorem of this system (without using axioms or deduction rules external to this system). This impossibility is a special case of a more general impossibility: no concept may define itself or compare itself to itself, since a definition or comparison requires an encompassing set.
Therefore, the consistency of an axiomatic system is certain only as long as no inconsistency has been discovered in it!
Conclusion: in spite of its rigor and elegance, an axiomatic approach has intrinsic limitations.
§ Level of truth achievable using the scientific method
Today's scientific method based on Karl Popper's critical rationalism, considers plausible every statement (formula, sentence or entire text) which:
· Is falsifiable, because it may be proven false by a theoretical demonstration or a single experimental counterexample (one will suffice);
· Has been examined by the scientific community and has not been refuted.
The level of truth achievable by this approach is not based on the number of experimental confirmations of the statement; it is based on a consensus of specialists on the impossibility to disprove it. As long as it has not been examined by other scientists, an author's proposal is only that: it is a suggestion, a conjecture, a hypothesis, a suggested proof of a theorem, or an account of experiments. A plausible statement becomes true if it has at least one theoretical or experimental proof that has been examined without being refuted.
A statement is always considered temporarily true, since it can be disproved tomorrow by a new discovery. A postulate (axiom) on which other statements have been based for some time without refutation has the same level of truth as an experimental (empirical) law without a single counterexample. The level of truth of a theorem, logically deduced from postulated axioms, is different since it is impossible to contradict a sequence of logical deductions without calling into question its base axioms.
With all this in mind, the issue for the author of a new discovery is obtaining the attention of the scientific community. This is often very difficult, because publication in respected media such as Physical Review or Nature is filtered by reading committees whose open-mindedness and neutrality are not necessarily always perfect. In addition, the limited amount of space of each issue creates a competition between texts and publication delays. Publication is easier in less famous media, on the Internet or through a publishing house, but without certainty of attracting enough attention to obtain in-depth reviews and criticisms. This problem is not new: Ludwig Boltzmann, the great physicist who developed statistical mechanics and whose work enabled the discoveries of Planck and Einstein, committed suicide in 1906 in part because his work had not attracted enough attention.
An additional problem is that research funds are often allocated by politicians whose motives are all but scientific; and sometimes fund allocation is subject to the approval of specialists who are competitors and would like to reserve the funds and potential fame to their own work…
Conclusion: what can we know?
For simplicity's sake, I consider two levels of knowledge of a field:
§ The in-depth comprehension of specialists, who can appreciate the field's hypotheses and limits and use it to anticipate evolutions of phenomena.
§ Superficial understanding, which allows situating the field among others and appreciating its relationships with them. This level of knowledge is suitable, for example, for philosophers who want to reflect beyond scientific knowledge, if they subsequently verify their conclusions in debates with specialists.
Nowadays, the level of abstraction and complexity of knowledge at specialist level make in-depth knowledge of a field a profession. A specialist must frequently exchange ideas and results with colleagues of the same field and related fields; he must also be very involved in his work and take it to heart. Some fields such as mathematics require specific intellectual qualities. But all of this is well known, my real interest here is superficial knowledge.
Knowing a subject only superficially is no excuse for believing in falsehoods or inventing answers to questions. An intellectual needs a fair amount of general knowledge that comprises – among others – a basic understanding of subjects discussed above, such as:
§ Determinism and randomness;
§ Concepts of physics such as discontinuity (quantization) and probabilistic variables;
§ Reliable reasoning methods such as hierarchical representations, axiomatic systems and critical rationalism.
The elements of such a culture are available today in texts, some of which are accessible for free on the Internet. Acquiring this culture requires more than reading time and a commitment to keep one's knowledge up-to-date, it requires some training in intellectual rigor. Too many people frequently think without rigor, for example:
§ Politicians, journalists and sales staff who base their thoughts on false arguments or illusions, and talk a lot of fine words;
§ Persons whose literary culture oriented towards fantasy and emotions discourages them to reflect seriously, and encourages them to follow their intuitions rather than strive to seek factual truth.
Unfortunately, in our society knowledge and reflecting are devalued for the benefit of entertainment, emotion, watching sports, and other play activities with little intellectual content. Nowadays we have more possibilities to know and understand than at any time in the past, but for too many people learning and thinking are not worth the effort.
Appendix
The fundamental hypothesis of philosophical determinism (the axiom of a single chain of events that starts infinitely far in the past and continues infinitely far in the future) is known as Laplace's determinism. In his book of 1814 "A Philosophical Essay on Probabilities", he wrote:
"We should consider the present state of the Universe as the effect of its previous state and the cause of the state that will follow. An intelligence which, at a given time, would know all of the forces that govern nature and the respective state of all its beings – assuming it is vast enough to analyze that data – would grasp in the same formula the movements of the largest bodies of the Universe and those of its lightest atom; nothing would be uncertain for it, the future and the past alike would stand before its eyes."
(That intelligence is often termed "Laplace's demon").
Two articles about non-separability
§ Nature 398, 189-190 (18 March 1999) - "Bell's inequality test: more ideal than ever" http://npg.nature.com/nature/journal/v398/n6724/full/398189a0.html;jsessionid=DE31F4383430A53DE45AD176658DC5F3
§ French Academy of Sciences - "Cette étonnante mécanique quantique", a speech by Alain Aspect (June 17, 2002) http://www.academie-sciences.fr/conferences/seances_solennelles/pdf/discours_Aspect_17_06_02.pdf