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In [[philosophy]] the term '''free will''' refers to consideration of whether an individual has the ability to make decisions or, alternatively, has only the illusion of doing so.  It is an age-old concern to separate what we can do something about, choose to do, from what we cannot.  The underlying quandary is the idea that science suggests future events are dictated to a great extent, and perhaps entirely, by past events and, inasmuch as the human body is part of the world science describes, its actions also are determined by physical laws and are not affected by human decisions. This view of events is a particular form of ''determinism'', sometimes called ''physical reductionism'',<ref name=Ney/>  and the view that determinism precludes free will is called ''incompatibilism''.<ref name=Vihvelin/> 


{{Image|High-pass amplifier Bode plot.PNG|right|300px|The Bode plot for a first-order (one-pole) [[highpass filter]]; the straight-line approximations are labeled "Bode pole"; phase varies from 90° at low frequencies (due to the contribution of the numerator, which is 90° at all
There are several ways to avoid the incompatibilst position, resulting in various ''compatibilist'' positions.<ref name=Timpe/>  One is to limit the scope of scientific description in a manner that excludes human decisions. Another is to argue that even if our actions are strictly determined by the past, it doesn’t seem that way to us, and so we have to find an approach to this issue that somehow marries our intuition of independence with the reality of its fictional nature. A third, somewhat legalistic approach, is to suggest that the ‘will’ to do something is quite different from actually doing it, so ‘free will’ can exist even though there may be no freedom of action.
frequencies) to 0° at high frequencies (where the phase contribution of the denominator is −90° and cancels the contribution of the numerator).}}
{{Image|Low-pass amplifier Bode plot.PNG|right|300px|The Bode plot for a first-order (one-pole) [[lowpass filter]]; the straight-line approximations are labeled "Bode pole"; phase is 90° lower than for the highpass filter because the phase contribution of the numerator is 0° at all frequencies.}}
A '''Bode plot''', named after [[Hendrik Wade Bode]], is a combination of a Bode magnitude plot and Bode phase plot:


A '''Bode magnitude plot''' is a graph of [[logarithm|log]] magnitude versus [[frequency]], plotted with a log-frequency axis, to show the [[transfer function]] or [[frequency response]] of a [[LTI system theory|linear, time-invariant]] system.  The magnitude axis of the Bode plot is usually expressed as [[decibel]]s, that is, 20 times the common logarithm of the amplitude of the gain.
There is also a theological version of the dilemma. roughly, if a deity or deities, or 'fate', controls our destiny, what place is left for free will?


A '''Bode phase plot''' is a graph of phase versus frequency, also plotted on a log-frequency axis,to evaluate how much a signal will be [[Phase (waves)|phase-shifted]]. The phase shift Φ is generally a function of frequency.
==Science does not apply==


The two Bode plots can be seen as separate plots of the real and the imaginary parts of the complex logarithm of a complex gain, say ''A''(&phi;) = |''A''|e<sup>j&phi;</sup>. The Bode plot of a gain that is the product of two gains, ''A = A<sub>1</sub>A<sub>2</sub>'' is thus:
One approach to limiting the applicability of science to our decisions is the examination of the notion of cause and effect.  For example, [[David Hume]] suggested that science did not really deal with causality, but with the correlation of events. So, for example, lighting a match in a certain environment does not ‘’cause’’ an explosion, but is ‘’associated’’ with an explosion. [[Immanuel Kant]] suggested that the idea of cause and effect is not a fact of nature but an interpretation put on events by the human mind, a ‘programming’ built into our brains.  Assuming this criticism to be true, there may exist classes of events that escape any attempt at cause and effect explanations.


:<math>\log \left( |A_1|e^{j\phi_1}\ \cdot \ |A_2|e^{j\phi_2}\right) </math>&emsp;<math>= \log \left(|A_1||A_2|\right) + \log e^{j(\phi_1+\phi_2)} </math>&emsp;<math>=\left(\log|A_1|+ \log|A_2|\right) +j(\phi_1+\phi_2)\ , </math>
A different way to exempt human decision from the scientific viewpoint is to note that science is a human enterprise.  It involves the human creation of theories to explain certain observations, and moreover, the observations it chooses to attempt to explain are selected, and do not encompass all experience.  For example, we choose to explain phenomena like the [[Higgs boson]] found by elaborate means like a [[hadron collider]], but don’t attempt to explain other phenomena that do not appear amenable to science at this time, often suggesting that they are beneath attention.
that is, the Bode plot of the ''product'' is expressed by the two Bode plots: the ''sum'' of the Bode magnitude plots and the ''sum'' of the Bode phase plots.  
As time progresses, one may choose to believe that science will explain all experience, but that view must be regarded as speculation analogous to predicting the stock market on the basis of past performance.  


In the figure at right, the Bode magnitude and phase plots are shown for a one-pole [[highpass filter]] function:
Although not explicitly addressing the issue of free will, it may be noted that [[Ludwig Wittgenstein]] argued that the specialized theories of science, as discussed by [[Rudolf Carnap]] for example, inevitably cover only a limited range of experience.  [[Stephen Hawking|Hawking/Mlodinow]] also noted this fact in in their [[model-dependent realism]],<ref name=Hawking/> the observation that, from the scientific viewpoint, reality is covered by a patchwork of theories that are sometimes disjoint and sometimes overlap.
{{quote|“Whatever might be the ultimate goals of some scientists, science, as it is currently practiced, depends on multiple overlapping descriptions of the world, each of which has a domain of applicability. In some cases this domain is very large, but in others quite small.”<ref name=Davies/>}}
::: —— E.B. Davies <span style="font-size:88%">''Epistemological pluralism'', p. 4</span>


::<math> \mathrm{A_{High}}(f) = \frac {j f/  f_1} {1 + j f/f_1} \ , </math>
Still another approach to this matter is analysis of the mind-brain connection (more generally, the [[mind-body problem]]). As suggested by Northoff,<ref name=Northoff/> there is an observer-observation issue involved here. Observing a third-person’s mental activity is a matter for neuroscience, perhaps strictly a question of neurons and their interactions through complex networks.  But observing our own mental activity is not possible in this way – it is a matter of subjective experiences.  The suggestion has been made that ‘’complementary’’ descriptions of nature are involved, that may be simply different perspectives upon the same reality:
{{quote|“...for each individual there is ''one'' 'mental life' but ''two'' ways of knowing it: first-person knowledge and third-person knowledge. From a first-person perspective conscious experiences appear causally effective. From a third person perspective the same causal sequence can be explained in neural terms. It is not the case that the view from one perspective is right and the other wrong. These perspectives are complementary. The differences between how things appear from a first-person versus a third-person perspective has to do with differences in the ''observational arrangements'' (the means by which a subject and an external observer access the subject's mental processes).”<ref name=Velmans/> |Max Velmans: |How could conscious experiences affect brains?, p. 11''
}}


where ''f'' is the frequency in Hz, and ''f''<sub>1</sub> is the pole position in Hz, ''f''<sub>1</sub> = 100 Hz in the figure. Using the rules for [[complex number]]s, the magnitude of this function is  
A related view is that the two descriptions may be mutually exclusive. That is, the connection between subjective experience and neuronal activity may run into a version of the measurement-observation interference noticed by [[Niels Bohr]] and by [[Erwin Schrödinger]] in the early days of quantum mechanics. (The measurement of the position of a particle caused the particle to change position in an unknown way.)
{{quote|“...it is important to be clear about exactly what experience one wants one's subjects to introspect. Of course, explaining to subjects exactly what the experimenter wants them to experience can bring its own problems–...instructions to attend to a particular internally generated experience can easily alter both the timing and he content of that experience and even whether or not it is consciously experienced at all.”<ref name=Pockett/> |Susan Pockett |The neuroscience of movement}}


::<math> \mid \mathrm{A_{High}}(f) \mid = \frac { f/f_1 } { \sqrt{ 1 + (f/f_1)^2 }} \ , </math>


while the phase is:
In any case, so far as free will is concerned, the implication of 'complementarity' is that 'free will' may be a description that is either an alternative to the scientific view, or possibly a view that can be entertained only if the scientific view is abandoned.


::<math> \phi_{A_{High}} = 90^\circ - \mathrm{ tan^{-1} } (f/f_1) \ . </math>
==Science can be accommodated==
A second approach is to argue that we can accommodate our subjective notions of free will with a deterministic reality. One way to do this is to argue that although we cannot do differently, in fact we really don’t want to do differently, and so what we ‘decide’ to do always agrees with what we (in fact) have to do.  Our subjective vision of the decision process as ‘voluntary’ is just a conscious concomitant of the unconscious and predetermined move to action.


The adjective ''highpass'' refers to the behavior that high frequency signals are transferred unchanged, while low frequency signals are attenuated.
==’Will’ ''versus'' ‘action’==
There is growing evidence of the pervasive nature of subconscious thought upon our actions, and the capriciousness of consciousness,<ref name=Norretranders/> which may switch focus from a sip of coffee to the writing of a philosophical exposition without warning. There also is mounting evidence that our consciousness is greatly affected by events in the brain beyond our control. For example, [[addiction|drug addiction]] has been related to alteration of the mechanisms in the brain for [[dopamine]] production, and withdrawal from addiction requires a reprogramming of this mechanism that is more than a simple act of will. The ‘will’ to overcome addiction can become separated from the ability to execute that will.
{{quote|“Philosophers who distinguish ''freedom of action'' and ''freedom of will'' do so because our success in carrying out our ends depends in part on factors wholly beyond our control. Furthermore, there are always external constraints on the range of options we can meaningfully try to undertake. As the presence or absence of these conditions and constraints are not (usually) our responsibility, it is plausible that the central loci of our responsibility are our choices, or ‘willings’.”[Italics not in original.]<ref name=OConnor/>| Timothy O'Connor |Free Will}}
In effect, could the 'will' be a subjective perception which might operate outside the realm of scientific principle, while its execution is not?


Care must be taken that the inverse tangent is set up to return ''degrees'', not radians. On the Bode magnitude plot, decibels are used, and the plotted magnitude is:
==Theology==
The ancient Greeks held the view that the gods ''could'' intervene in the course of events, and it was possible on occasion to divine their intentions or even to change them. That view leaves a role for free will, although it can be limited in scope by the gods. A more complete restriction is the belief that the gods are omniscient and have perfect foreknowledge of events, which obviously includes human decisions. This view leads to the belief that, while the gods know what we will choose, humans do not, and are faced therefore with playing the role of deciding our actions, even though they are scripted, a view contradicted by Cassius in arguing with Brutus, a Stoic:
{{quote| “The fault, dear Brutus, is not in our stars, But in ourselves, that we are underlings.”|spoken by Cassius| Julius Caesar (I, ii, 140-141)}}
The [[Stoicism|Stoics]] wrestled with this problem, and one argument for compatibility took the view that although the gods controlled matters, what they did was understandable using human intellect. Hence, when fate presented us with an issue, there was a duty to sort through a decision, and assent to it (a ''responsibility''), a sequence demanded by our natures as rational beings.<ref name=Bobzien/>


:<math>20\ \mathrm{log_{10}} \mid \mathrm{A_{High}}(f) \mid \ =20\  \mathrm{log_{10}} \left( f/f_1 \right)</math>
In [http://www.iep.utm.edu/chrysipp/ Chrysippus of Soli's] view (an apologist for Stoicism), ''fate'' precipitates an event, but our nature determines its course, in the same way that bumping a cylinder or a cone causes it to move, but it rolls or it spins according to its nature.<ref name=Bobzien2/> The actual course of events depends upon the nature of the individual, who therefore bears a personal responsibility for the resulting events. It is not clear whether the individual is thought to have any control over their nature, or even whether this question has any bearing upon their responsibility.<ref name=Bobzien3/>
:::::::&emsp; <math>\ -20  \  \mathrm{log_{10}} \left( \sqrt{ 1 + (f/f_1)^2 }\right) \ . </math>
In the next figure, the Bode plots are shown for the one-pole [[lowpass filter]] function:


::<math> \mathrm{ A_{Low}} (f) = \frac {1} {1 + j f/f_1} \ . </math>  
==References==
{{reflist|refs=
<ref name=Bobzien>
{{cite book |author=Susanne Bobzien |title=Determinism and Freedom in Stoic Philosophy |url=http://books.google.com/books?id=7kmTeOjHIqkC&printsec=frontcover |year=1998 |publisher=Oxford University Press |isbn=0198237944}} See §6.3.3 ''The cylinder and cone analogy'',  pp. 258 ''ff''.  
</ref>


The ''lowpass'' adjective describes the unattenuated passage of low-frequency signals and the attenuation of high-frequency signals.  
<ref name=Bobzien2>
{{cite book |author=Susanne Bobzien |title=Determinism and Freedom in Stoic Philosophy |url=http://books.google.com/books?id=7kmTeOjHIqkC&printsec=frontcover |year=1998 |publisher=Oxford University Press |isbn=0198237944}} See in particular pp. 386 ''ff''.  
</ref>


Circuit modifications seldom change the magnitude and phase Bode plots independently — changing the amplitude response of the system will most likely change the phase characteristics and ''vice versa''. Their interdependence is illustrated by the observation that the phase and amplitude characteristics can be obtained from each other for minimum-phase systems using the [[Hilbert transform]].
<ref name=Bobzien3>
{{cite book |author=Susanne Bobzien |title=Determinism and Freedom in Stoic Philosophy |url=http://books.google.com/books?id=7kmTeOjHIqkC&printsec=frontcover |year=1998 |publisher=Oxford University Press |isbn=0198237944}} See in particular p. 255.  
</ref>


If the transfer function is a [[rational function]] with real poles and zeros, then the Bode plot can be approximated with straight lines. These asymptotic approximations are called '''straight line Bode plots'''. Also shown in these two figures are these straight-line approximations to the Bode plots.


==An example with pole and zero==
<ref name=Davies>
{{Image|Bode plot for pole and zero.PNG|right|350px| Bode magnitude plot for zero and for low-pass pole; curves labeled "Bode" are the straight-line Bode plots.}}
{{cite web |title=Epistemological pluralism |author=E Brian Davies |url=http://philsci-archive.pitt.edu/3083/1/EP3single.doc |work=PhilSci Archive |year=2006 }}
{{Image|Bode phase plot for pole and zero.PNG|right|350px| Bode phase plot for zero and for low-pass pole; curves labeled "Bode" are the straight-line Bode plots.}}
</ref>
{{Image|Superposed Bode plots for pole and zero.PNG|right|350px|Bode magnitude plot for pole-zero combination; the location of the zero is ten times higher than in above figures; curves labeled "Bode" are the straight-line Bode plots}}
{{Image|Superposed Bode phase plots for pole and zero.PNG|right|350px| Bode phase plot for pole-zero combination; the location of the zero is ten times higher than in above figures; curves labeled "Bode" are the straight-line Bode plots.}}
The construction of Bode plots using superposition now is illustrated. To begin, the components are presented separately.


The figure at right shows the Bode magnitude plot for a zero and a low-pass pole, and compares the two with the Bode straight line plots. The straight-line plots are horizontal up to the pole (zero) location and then drop (rise) at 20 dB/decade. The figure below does the same for the phase. The phase plots are horizontal up to a frequency a factor of ten below the pole (zero) location and then drop (rise) at 45°/decade until the frequency is ten times higher than the pole (zero) location. The plots then are again horizontal at higher frequencies at a final, total phase change of 90°.


The Bode plot for a gain function that is the product of a pole and zero can be constructed by superposition, because the Bode plot is logarithmic, and the logarithm of a product of factors is sum of the individual, separate logarithms. The following two figures show how superposition (simple addition) of a pole and zero plot is done. The Bode straight line plots again are compared with the exact plots. The zero is assumed to reside at higher frequency than the pole to make a more interesting example.
<ref name=Hawking>
{{cite book |author=Hawking SW, Mlodinow L. |title=The Grand Design |isbn=0553805371 |url= http://www.amazon.com/Grand-Design-Stephen-Hawking/dp/0553805371#reader_0553805371 |pages=pp. 42-43 |chapter=Chapter 3: What is reality?|year=2010|publisher=Bantam Books}}
</ref>


Notice in the magnitude plot that the initial 20 dB/decade drop of the pole is arrested by the onset of the 20 dB/decade rise of the zero, resulting in a ''horizontal'' (zero-slope) magnitude plot for frequencies above the zero location.
<ref name=Norretranders>
{{cite book |url=http://www.google.com/search?tbo=p&tbm=bks&q=consciousness%2Bplays%2Ba%2Bsmaller%2Brole%2Bin%2Bhuman%2Blife+intitle:User+intitle:illusion&num=10 |title=The user illusion: Cutting consciousness down to size |quote=Consciousness plays a far smaller role in human life than Western culture has tended to believe |author=Tor Nørretranders |isbn=0140230122 |chapter=Preface |pages=p. ''ix'' |publisher=Penguin Books |year=1998 |edition=Jonathan Sydenham translation of ''Maerk verden'' 1991 ed }}
</ref>


Notice in the bottom phase plot that the straight-line approximation is very approximate in the region where both pole and zero affect the phase. Notice also in this plot that the range of frequencies where the phase changes in the straight line plot is limited to frequencies a factor of ten above and below the pole (zero) location. Where the phase of the pole and the zero both are present, the straight-line phase plot is horizontal because the 45°/decade drop of the pole is arrested by the overlapping 45°/decade rise of the zero in the limited range of frequencies where both are active contributors to the phase.
<ref name=Ney>
{{cite web |author=Alyssa Ney |title=Reductionism |work=Internet Encyclopedia of Philosophy |date= November 10, 2008 |url=http://www.iep.utm.edu/red-ism/}}
</ref>


==Gain margin and phase margin==
<ref name=Northoff>
Bode plots are used to assess the stability of negative feedback amplifiers by finding the gain and [[phase margin]]s of an amplifier. The notion of gain and phase margin is based upon the gain expression for a [[negative feedback amplifier]] given by
A rather extended discussion is provided in {{cite book |title=Philosophy of the Brain: The Brain Problem |author=Georg Northoff |url=http://books.google.com/books?id=r0Bf3lLys6AC&printsec=frontcover |publisher=John Benjamins Publishing |isbn=1588114171 |year=2004 |edition=Volume 52 of Advances in Consciousness Research}}
 
</ref>
::<math> A_{FB} = \frac {A_{OL}} {1 + \beta A_{OL}} \ , </math>
 
where A<sub>FB</sub> is the gain of the amplifier with feedback (the '''closed-loop gain'''), β is the '''feedback factor''' and ''A''<sub>OL</sub> is the gain without feedback (the '''open-loop gain'''). The gain ''A''<sub>OL</sub> is a complex function of frequency, with both magnitude and phase.<ref name=note1/> Examination of this relation shows the possibility of infinite gain (interpreted as instability) if the product β''A''<sub>OL</sub> = −1. (That is, the magnitude of β''A''<sub>OL</sub> is unity and its phase is −180°, the so-called '''Barkhausen  criteria'''). Bode plots are used to determine just how close an amplifier comes to satisfying this condition.
 
Key to this determination are two frequencies. The first, labeled here as ''f''<sub>180</sub>, is the frequency where the open-loop gain flips sign. The second, labeled here ''f''<sub>0dB</sub>, is the frequency where the magnitude of the product | β ''A''<sub>OL</sub> | = 1 (in dB, magnitude 1 is 0 dB). That is, frequency ''f''<sub>180</sub> is determined by the condition:
 
:::<math> \beta A_{OL} \left( f_{180} \right) = - | \beta A_{OL} \left( f_{180} \right)| = - | \beta A_{OL}|_{180} \ , </math>
 
where vertical bars denote the magnitude of a complex number (for example, | a + j b | = [ a<sup>2</sup> + b<sup>2</sup>]<sup>1/2</sup> ), and frequency ''f''<sub>0dB</sub> is determined by the condition:
 
:::<math>| \beta A_{OL} \left( f_{0dB} \right) | = 1 \ . </math>
 
One measure of proximity to instability is the '''gain margin'''. The Bode phase plot locates the frequency where the phase of β''A''<sub>OL</sub> reaches −180°, denoted here as frequency ''f''<sub>180</sub>. Using this frequency, the Bode magnitude plot finds the magnitude of β''A''<sub>OL</sub>. If |β''A''<sub>OL</sub>|<sub>180</sub> = 1, the amplifier is unstable, as mentioned. If |β''A''<sub>OL</sub>|<sub>180</sub> < 1, instability does not occur, and the separation in dB of the magnitude of |β''A''<sub>OL</sub>|<sub>180</sub> from |β''A''<sub>OL</sub>| = 1 is called the ''gain margin''. Because a magnitude of one is 0 dB, the gain margin is simply one of the equivalent forms: 20 log<sub>10</sub>( |β''A''<sub>OL</sub>|<sub>180</sub>) = 20 log<sub>10</sub>( |''A''<sub>OL</sub>|<sub>180</sub>) − 20 log<sub>10</sub>( 1 / β ).  
 
Another equivalent measure of proximity to instability is the '''[[phase margin]]'''. The Bode magnitude plot locates the frequency where the magnitude of |β''A''<sub>OL</sub>| reaches unity, denoted here as frequency ''f''<sub>0dB</sub>. Using this frequency, the Bode phase plot finds the phase of β''A''<sub>OL</sub>. If the phase of β''A''<sub>OL</sub>( ''f''<sub>0dB</sub>) > −180°,  the instability condition cannot be met at any frequency (because its magnitude is going to be < 1 when ''f = f''<sub>180</sub>), and the distance of the phase at ''f''<sub>0dB</sub> in degrees above −180° is called the ''phase margin''.
 
If a simple ''yes'' or ''no'' on the stability issue is all that is needed, the amplifier is stable if ''f''<sub>0dB</sub> < ''f''<sub>180</sub>. This criterion is sufficient to predict stability only for amplifiers satisfying some restrictions on their pole and zero positions ([[minimum phase]] systems). Although these restrictions usually are met, if they are not another method must be used, such as the [[Nyquist plot]].<ref name=Lee/><ref name=Levine/>
{{Image|Open and closed loop gain.PNG|right|350px| Gain of feedback amplifier ''A''<sub>FB</sub> in dB and corresponding open-loop amplifier ''A''<sub>OL</sub>. The gain margin in this amplifier is nearly zero because {{nowrap|β''A''<sub>OL</sub> <nowiki>=</nowiki> 1}} occurs at almost {{nowrap|''f'' <nowiki>=</nowiki> ''f''<sub>180°</sub>.}}}}
{{Image|Open and closed loop phase.PNG|right|350px|Phase of feedback amplifier ''°A''<sub>FB</sub> in degrees and corresponding open-loop amplifier ''°A''<sub>OL</sub>. The phase margin in this amplifier is nearly zero because the phase-flip occurs at almost the unity gain frequency ''f'' <nowiki>=</nowiki> ''f''<sub>0dB</sub> where {{nowrap|β''A''<sub>OL</sub> <nowiki>=</nowiki> 1.}}}}
 
===Examples using Bode plots===
Two examples illustrate gain behavior and terminology. For a three-pole amplifier, gain and phase plots for a borderline stable and a stable amplifier are compared.
 
====Borderline stable amplifier====
The first of two figures at the right compares the Bode plots for the gain without feedback (the ''open-loop'' gain) ''A''<sub>OL</sub> with the gain with feedback ''A''<sub>FB</sub> (the ''closed-loop'' gain). See [[negative feedback amplifier]] for more detail.
 
Because the open-loop gain ''A''<sub>OL</sub> is plotted and not the product β ''A''<sub>OL</sub>, the condition {{nowrap|''A''<sub>OL</sub> = 1 / β}} decides the frequency where {{nowrap|''&beta;A<sub>OL</sub>'' = 1,}} that is the frequency labeled ''f''<sub>0dB</sub>. The feedback gain at low frequencies and for large ''A''<sub>OL</sub> is {{nowrap|''A''<sub>FB</sub> &asymp; 1 / β}} (look at the formula for the feedback gain at the beginning of this section for the case of large gain ''A''<sub>OL</sub>), so an equivalent way to find ''f''<sub>0dB</sub> is to look where the feedback (or closed-loop) gain intersects the open-loop gain. (Frequency ''f''<sub>0dB</sub> is needed later to find the phase margin.)
 
Near the crossover of the two gains at ''f''<sub>0dB</sub>, the Barkhausen criteria are almost satisfied in this example, and the feedback amplifier exhibits a massive peak in gain (it would be infinity if {{nowrap|β ''A''<sub>OL</sub> = −1).}} Beyond the unity gain frequency ''f''<sub>0dB</sub>, the open-loop gain is sufficiently small that {{nowrap|''A''<sub>FB</sub> ≈ ''A''<sub>OL</sub>}} (examine the formula at the beginning of this section for the case of small ''A''<sub>OL</sub>).
 
The second of the two figures shows the corresponding phase comparison: the phase of the feedback amplifier is nearly zero out to the frequency ''f''<sub>180</sub> where the open-loop gain has a phase of −180°. In this vicinity, the phase of the feedback amplifier plunges abruptly downward to become almost the same as the phase of the open-loop amplifier. (Recall, {{nowrap|''A''<sub>FB</sub> ≈ ''A''<sub>OL</sub>}} for small ''A''<sub>OL</sub>.)
 
Comparing the labeled points in these two figures, it is seen that the unity gain frequency ''f''<sub>0dB</sub> and the phase-flip frequency ''f''<sub>180</sub> are very nearly equal in this amplifier, ''f''<sub>180</sub> &asymp; ''f''<sub>0dB</sub> &asymp; 3.332 kHz, which means the gain margin and phase margin are nearly zero. The amplifier is borderline stable.
 
====Stable example====
{{Image|Gain margin.PNG|right|350px|Gain of feedback amplifier ''A''<sub>FB</sub> in dB and corresponding open-loop amplifier ''A''<sub>OL</sub>. The gain margin in this amplifier is 19 dB.}}
{{Image|Phase margin.PNG|right|350px|Phase of feedback amplifier ''A''<sub>FB</sub> in degrees and corresponding open-loop amplifier ''A''<sub>OL</sub>. The phase margin in this amplifier is 45°.}}
The last two figures on the right illustrate the gain margin and phase margin for a different amount of feedback β. The feedback factor is chosen smaller than in previous borderline stable amplifier, moving the the condition {{nowrap|<nowiki>|</nowiki> β ''A''<sub>OL</sub> <nowiki>|</nowiki> <nowiki>=</nowiki> 1}} to the lower frequency of f<sub>0dB</sub> = 1 kHz.
 
The upper of the two figures shows the gain plot. The intersection of {{nowrap|1 / β}} and ''A''<sub>OL</sub> occurs at  {{nowrap|''f''<sub>0dB</sub> <nowiki>=</nowiki> 1 kHz.}} Notice that the peak in the gain ''A''<sub>FB</sub> near ''f''<sub>0dB</sub> seen in the borderline stable amplifier is almost gone.<ref name=note2/><ref name=Sansen/>
The lower of the two figures is the phase plot. Using the value of ''f''<sub>0dB</sub> = 1 kHz found above from the magnitude plot, the open-loop phase at ''f''<sub>0dB</sub> is −135°, which is a phase margin of 45° above −180°.
 
Using the phase plot, for a phase of −180° the value of {{nowrap|''f''<sub>180</sub> <nowiki>=</nowiki> 3.332 kHz}} (the same result as found earlier, of course<ref name=note3/>). The open-loop gain from the gain plot at ''f''<sub>180</sub> is 58 dB, and {{nowrap|1 / β <nowiki>=</nowiki> 77 dB,}} so the gain margin is 19 dB.
 
As an aside, it should be noted that stability is not the sole criterion for amplifier response, and in many applications a more stringent demand than stability is good [[Step_response#Step_response_of_feedback_amplifiers|step response]]. As a [[rule of thumb]], good step response requires a phase margin of at least 45°, and often a margin of over 70° is advocated, particularly where component variation due to manufacturing tolerances is an issue.<ref name=Sansen2/> See also the discussion of phase margin in the [[Step_response#Phase_margin|step response]] article.


==References and notes==
<ref name=OConnor>
{{reflist|refs=
{{cite web |title=&thinsp;Free Will |date=Oct 29, 2010 |author=O'Connor, Timothy |url=http://plato.stanford.edu/archives/sum2011/entries/freewill |work=The Stanford Encyclopedia of Philosophy (Summer 2011 Edition) |editor=Edward N. Zalta, ed.}}
<ref name=note1>
Ordinarily, as frequency increases the magnitude of the gain drops and the phase becomes more negative, although these are only trends and may be reversed in particular frequency ranges. Unusual gain behavior can render the concepts of gain and phase margin inapplicable. Then other methods such as the [[Nyquist plot]] have to be used to assess stability.
</ref>
</ref>


<ref name=Lee>
<ref name=Pockett>
{{cite book  
{{cite book|title=&thinsp;Does Consciousness Cause Behavior? |chapter=The neuroscience of movement |author=Susan Pockett |url=http://books.google.com/books?id=G5CaTnNksgkC&pg=PA19&lpg=PA19 |pages= p. 19 |editor=Susan Pockett, WP Banks, Shaun Gallagher, eds.  |publisher=MIT Press |date =2009 |isbn=0262512572}}
|author=Thomas H. Lee
|title=The design of CMOS radio-frequency integrated circuits
|page=§14.6 pp. 451-453
|year= 2004
|edition=Second Edition
|publisher=Cambridge University Press
|location=Cambridge UK
|isbn=0-521-83539-9
|url=http://worldcat.org/isbn/0-521-83539-9}}
</ref>
</ref>


<ref name=Levine>
<ref name=Timpe>
{{cite book
{{cite web |author=Kevin Timpe |title=Free will |work=Internet Encyclopedia of Philosophy |date= March 31, 2006 |url=http://www.iep.utm.edu/freewill/#H5}}
|author=William S Levine
|title=The control handbook: the electrical engineering handbook series
|page=§10.1 p. 163
|year= 1996
|edition=Second Edition
|publisher=CRC Press/IEEE Press
|location=Boca Raton FL
|isbn=0849385709
|url=http://books.google.com/books?id=2WQP5JGaJOgC&pg=RA1-PA163&lpg=RA1-PA163&dq=stability+%22minimum+phase%22&source=web&ots=P3fFTcyfzM&sig=ad5DJ7EvVm6In_zhI0MlF_6vHDA}}
</ref>
 
<ref name=note2>The critical amount of feedback where the peak in the gain ''just'' disappears altogether is the ''maximally flat'' or [[Butterworth_filter#Maximal_flatness|Butterworth]] design.
</ref>
</ref>


<ref name=note3>
The frequency where the open-loop gain flips sign ''f''<sub>180</sub> does not change with a change in feedback factor; it is a property of the open-loop gain. The value of the gain at ''f''<sub>180</sub> also does not change with a change in β. Therefore, we could use the previous values found for the borderline stable amplifier. However, for clarity the procedure is described using only the curves for the stable amplifier.
</ref>


<ref name=Sansen>
<ref name=Velmans>
{{cite book
{{cite journal |journal=Journal of Consciousness Studies |volume=9 |issue=11 |year=2002 |pages=pp. 2-29 |author=Max Velmans  |title=How Could Conscious Experiences Affect Brains? |url=http://cogprints.org/2750/ |year=2002}}
|author=Willy M C Sansen
|title=Analog design essentials
|page=§0517-§0527 pp. 157-163
|year= 2006
|publisher=Springer
|location=Dordrecht, The Netherlands
|isbn=0-387-25746-2
|url=http://worldcat.org/isbn/0-387-25746-2}}
</ref>
</ref>


 
<ref name=Vihvelin>
<ref name=Sansen2>
{{cite web |author=Kadri Vihvelin |title=&thinsp;Arguments for Incompatibilism |work=The Stanford Encyclopedia of Philosophy (Spring 2011 Edition) |editor=Edward N. Zalta, ed. |url= http://plato.stanford.edu/archives/spr2011/entries/incompatibilism-arguments/ |date=Mar 1, 2011}}
{{cite book
|author=Willy M C Sansen
|title=§0526 p. 162
|isbn=0-387-25746-2
|url=http://worldcat.org/isbn/0-387-25746-2}}
</ref>
</ref>


}}
}}

Latest revision as of 03:07, 22 November 2023


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In philosophy the term free will refers to consideration of whether an individual has the ability to make decisions or, alternatively, has only the illusion of doing so. It is an age-old concern to separate what we can do something about, choose to do, from what we cannot. The underlying quandary is the idea that science suggests future events are dictated to a great extent, and perhaps entirely, by past events and, inasmuch as the human body is part of the world science describes, its actions also are determined by physical laws and are not affected by human decisions. This view of events is a particular form of determinism, sometimes called physical reductionism,[1] and the view that determinism precludes free will is called incompatibilism.[2]

There are several ways to avoid the incompatibilst position, resulting in various compatibilist positions.[3] One is to limit the scope of scientific description in a manner that excludes human decisions. Another is to argue that even if our actions are strictly determined by the past, it doesn’t seem that way to us, and so we have to find an approach to this issue that somehow marries our intuition of independence with the reality of its fictional nature. A third, somewhat legalistic approach, is to suggest that the ‘will’ to do something is quite different from actually doing it, so ‘free will’ can exist even though there may be no freedom of action.

There is also a theological version of the dilemma. roughly, if a deity or deities, or 'fate', controls our destiny, what place is left for free will?

Science does not apply

One approach to limiting the applicability of science to our decisions is the examination of the notion of cause and effect. For example, David Hume suggested that science did not really deal with causality, but with the correlation of events. So, for example, lighting a match in a certain environment does not ‘’cause’’ an explosion, but is ‘’associated’’ with an explosion. Immanuel Kant suggested that the idea of cause and effect is not a fact of nature but an interpretation put on events by the human mind, a ‘programming’ built into our brains. Assuming this criticism to be true, there may exist classes of events that escape any attempt at cause and effect explanations.

A different way to exempt human decision from the scientific viewpoint is to note that science is a human enterprise. It involves the human creation of theories to explain certain observations, and moreover, the observations it chooses to attempt to explain are selected, and do not encompass all experience. For example, we choose to explain phenomena like the Higgs boson found by elaborate means like a hadron collider, but don’t attempt to explain other phenomena that do not appear amenable to science at this time, often suggesting that they are beneath attention. As time progresses, one may choose to believe that science will explain all experience, but that view must be regarded as speculation analogous to predicting the stock market on the basis of past performance.

Although not explicitly addressing the issue of free will, it may be noted that Ludwig Wittgenstein argued that the specialized theories of science, as discussed by Rudolf Carnap for example, inevitably cover only a limited range of experience. Hawking/Mlodinow also noted this fact in in their model-dependent realism,[4] the observation that, from the scientific viewpoint, reality is covered by a patchwork of theories that are sometimes disjoint and sometimes overlap.

‘ “Whatever might be the ultimate goals of some scientists, science, as it is currently practiced, depends on multiple overlapping descriptions of the world, each of which has a domain of applicability. In some cases this domain is very large, but in others quite small.”[5] ’
—— E.B. Davies Epistemological pluralism, p. 4

Still another approach to this matter is analysis of the mind-brain connection (more generally, the mind-body problem). As suggested by Northoff,[6] there is an observer-observation issue involved here. Observing a third-person’s mental activity is a matter for neuroscience, perhaps strictly a question of neurons and their interactions through complex networks. But observing our own mental activity is not possible in this way – it is a matter of subjective experiences. The suggestion has been made that ‘’complementary’’ descriptions of nature are involved, that may be simply different perspectives upon the same reality:

‘ “...for each individual there is one 'mental life' but two ways of knowing it: first-person knowledge and third-person knowledge. From a first-person perspective conscious experiences appear causally effective. From a third person perspective the same causal sequence can be explained in neural terms. It is not the case that the view from one perspective is right and the other wrong. These perspectives are complementary. The differences between how things appear from a first-person versus a third-person perspective has to do with differences in the observational arrangements (the means by which a subject and an external observer access the subject's mental processes).”[7] ’

—Max Velmans: , How could conscious experiences affect brains?, p. 11

A related view is that the two descriptions may be mutually exclusive. That is, the connection between subjective experience and neuronal activity may run into a version of the measurement-observation interference noticed by Niels Bohr and by Erwin Schrödinger in the early days of quantum mechanics. (The measurement of the position of a particle caused the particle to change position in an unknown way.)

‘ “...it is important to be clear about exactly what experience one wants one's subjects to introspect. Of course, explaining to subjects exactly what the experimenter wants them to experience can bring its own problems–...instructions to attend to a particular internally generated experience can easily alter both the timing and he content of that experience and even whether or not it is consciously experienced at all.”[8] ’

—Susan Pockett , The neuroscience of movement


In any case, so far as free will is concerned, the implication of 'complementarity' is that 'free will' may be a description that is either an alternative to the scientific view, or possibly a view that can be entertained only if the scientific view is abandoned.

Science can be accommodated

A second approach is to argue that we can accommodate our subjective notions of free will with a deterministic reality. One way to do this is to argue that although we cannot do differently, in fact we really don’t want to do differently, and so what we ‘decide’ to do always agrees with what we (in fact) have to do. Our subjective vision of the decision process as ‘voluntary’ is just a conscious concomitant of the unconscious and predetermined move to action.

’Will’ versus ‘action’

There is growing evidence of the pervasive nature of subconscious thought upon our actions, and the capriciousness of consciousness,[9] which may switch focus from a sip of coffee to the writing of a philosophical exposition without warning. There also is mounting evidence that our consciousness is greatly affected by events in the brain beyond our control. For example, drug addiction has been related to alteration of the mechanisms in the brain for dopamine production, and withdrawal from addiction requires a reprogramming of this mechanism that is more than a simple act of will. The ‘will’ to overcome addiction can become separated from the ability to execute that will.

‘ “Philosophers who distinguish freedom of action and freedom of will do so because our success in carrying out our ends depends in part on factors wholly beyond our control. Furthermore, there are always external constraints on the range of options we can meaningfully try to undertake. As the presence or absence of these conditions and constraints are not (usually) our responsibility, it is plausible that the central loci of our responsibility are our choices, or ‘willings’.”[Italics not in original.][10] ’

— Timothy O'Connor , Free Will

In effect, could the 'will' be a subjective perception which might operate outside the realm of scientific principle, while its execution is not?

Theology

The ancient Greeks held the view that the gods could intervene in the course of events, and it was possible on occasion to divine their intentions or even to change them. That view leaves a role for free will, although it can be limited in scope by the gods. A more complete restriction is the belief that the gods are omniscient and have perfect foreknowledge of events, which obviously includes human decisions. This view leads to the belief that, while the gods know what we will choose, humans do not, and are faced therefore with playing the role of deciding our actions, even though they are scripted, a view contradicted by Cassius in arguing with Brutus, a Stoic:

‘ “The fault, dear Brutus, is not in our stars, But in ourselves, that we are underlings.” ’

—spoken by Cassius, Julius Caesar (I, ii, 140-141)

The Stoics wrestled with this problem, and one argument for compatibility took the view that although the gods controlled matters, what they did was understandable using human intellect. Hence, when fate presented us with an issue, there was a duty to sort through a decision, and assent to it (a responsibility), a sequence demanded by our natures as rational beings.[11]

In Chrysippus of Soli's view (an apologist for Stoicism), fate precipitates an event, but our nature determines its course, in the same way that bumping a cylinder or a cone causes it to move, but it rolls or it spins according to its nature.[12] The actual course of events depends upon the nature of the individual, who therefore bears a personal responsibility for the resulting events. It is not clear whether the individual is thought to have any control over their nature, or even whether this question has any bearing upon their responsibility.[13]

References

  1. ↑ Alyssa Ney (November 10, 2008). Reductionism. Internet Encyclopedia of Philosophy.
  2. ↑ Kadri Vihvelin (Mar 1, 2011). Edward N. Zalta, ed.: Arguments for Incompatibilism. The Stanford Encyclopedia of Philosophy (Spring 2011 Edition).
  3. ↑ Kevin Timpe (March 31, 2006). Free will. Internet Encyclopedia of Philosophy.
  4. ↑ Hawking SW, Mlodinow L. (2010). “Chapter 3: What is reality?”, The Grand Design. Bantam Books, pp. 42-43. ISBN 0553805371. 
  5. ↑ E Brian Davies (2006). Epistemological pluralism. PhilSci Archive.
  6. ↑ A rather extended discussion is provided in Georg Northoff (2004). Philosophy of the Brain: The Brain Problem, Volume 52 of Advances in Consciousness Research. John Benjamins Publishing. ISBN 1588114171. 
  7. ↑ Max Velmans (2002). "How Could Conscious Experiences Affect Brains?". Journal of Consciousness Studies 9 (11): pp. 2-29.
  8. ↑ Susan Pockett (2009). “The neuroscience of movement”, Susan Pockett, WP Banks, Shaun Gallagher, eds.:  Does Consciousness Cause Behavior?. MIT Press, p. 19. ISBN 0262512572. 
  9. ↑ Tor Nørretranders (1998). “Preface”, The user illusion: Cutting consciousness down to size, Jonathan Sydenham translation of Maerk verden 1991 ed. Penguin Books, p. ix. ISBN 0140230122. “Consciousness plays a far smaller role in human life than Western culture has tended to believe” 
  10. ↑ O'Connor, Timothy (Oct 29, 2010). Edward N. Zalta, ed.: Free Will. The Stanford Encyclopedia of Philosophy (Summer 2011 Edition).
  11. ↑ Susanne Bobzien (1998). Determinism and Freedom in Stoic Philosophy. Oxford University Press. ISBN 0198237944.  See §6.3.3 The cylinder and cone analogy, pp. 258 ff.
  12. ↑ Susanne Bobzien (1998). Determinism and Freedom in Stoic Philosophy. Oxford University Press. ISBN 0198237944.  See in particular pp. 386 ff.
  13. ↑ Susanne Bobzien (1998). Determinism and Freedom in Stoic Philosophy. Oxford University Press. ISBN 0198237944.  See in particular p. 255.
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