Rainer et al. (1999) ‘Prospective Coding for Objects in Primate Prefrontal Cortex’, notes

by Corry Shores
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Gregor Rainer; S. Chenchal Rao; and Earl K. Miller

Prospective Coding for Objects in Primate Prefrontal Cortex

Brief Summary:

This experiment suggests that when we retain visual information, it could be about the prior image we saw and/or about the forthcoming image we anticipate as associating with the prior one.

Notes:

In the introduction, the authors distinguish remembering and anticipating.

When a delay is imposed between two behaviorally related events, animals can bridge this gap by remembering the event before the delay or anticipating the event after the delay.
[p.5493]

They have us consider if the animal is shown two things [lets say red and blue dots] then later must choose one or another thing [for example, if see red, then delay, then must choose circle, or if see blue, then delay, then must choose square.] The animal can accomplish this either by retaining the image of the stimulus [the red or the blue dots] or by having in their memory an image of what they will need to choose [the circle or the square.] The case of remembering the prior image is retrospective processing, and the anticipation of the forthcoming image is prospective processing.

Consider a visual delayed paired associate (DPA) task in which there are two sample objects (e.g., “S1” and “S2”) and two choice objects (“C1” and “C2”). If S1 is the sample, the animal must choose C1 after the delay, and if S2 is the sample, the animal must choose C2. Animals could solve this task by maintaining over the delay a memory of the samples (S1 or S2) or a memory of the anticipated choice stimuli (C1 or C2). The former (memories of events in the recent past) is called retrospective processing, and the latter (anticipation of future events) is called prospective processing (Roitblat, 1980; Honig and Thompson, 1982).

Prospective processing allows animals to optimize behavior by preparing in advance for critical events. They can, for example, act more quickly and direct attention to catch a fleeting event that might otherwise be missed.
[p.5493]

In the experiment, two monkeys had sensors implanted to measure activity of certain neurons in the lateral prefrontal cortex (PF).

Some terms:

DPA: delayed paired associate task [like the S1/S2, C1/C2 task mentioned above.]

DMS: delayed match-to-sample [both the stimulus and the desired choice are the same image.]

Procedure:

Monkeys performed in alternate blocks DPA and DMS tasks.

(1) Monkey begins by grasping metal bar and fixate eyes on small spot for 2s.

(2) Then, a sample object is presented at the center of the gaze for 0.5s.

(3) Then there is a 1s delay.

(4) Then test object is displayed for 0.5s. If the test object matches the sample, the monkey is to release the metal bar within 0.5s in order to receive a juice reward. For DPA trials, there were prior months of training to associate the proper images.

Results:

There was increased activity in the neurons during delay periods. That activity increased as the test object appearance drew nearer.

Note that there is a marked increase in activity over the delay as the time of test object presentation approaches. Such “climbing activity” has been suggested to reflect prospective coding (Quintana and Fuster, 1992; Miller et al., 1996). [...] Although the monkey could not predict which sample would appear, it could predict when it would appear. This activity may reflect that expectation.
[p.5495]

Gregor Rainer; S. Chenchal Rao; and Earl K. Miller. Prospective Coding for Objects in Primate Prefrontal Cortex. The Journal of Neuroscience, 1 July 1999, 19(13): 5493-5505.

http://www.jneurosci.org.sci-hub.org/content/19/13/5493.short

https://www.earlkmiller.org/wp-content/uploads/2013/03/Rainer-Journal-of-Neuroscience-1999.pdf

Miller et al. (1996) ‘Neural Mechanisms of Visual Working Memory in Prefrontal Cortex of the Macaque’, notes

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Earl K. Miller; Cynthia A. Erickson;and Robert Desimone

Neural Mechanisms of Visual Working Memory in Prefrontal Cortex of the Macaque

Brief summary:

This research suggests that our brains retain visual information from the past by continuing neural activity in the present.

Notes:

Inserted electrodes monitored the activity of neurons in the prefrontal cortex of two monkeys’ brains.

Procedure:

(1) Monkey grabs small bar and fixates on spot of light at the screen’s center (target fixation). The monkey must maintain fixation for 300msec before the first stimulus is presented.

(2) One of the images was shown [does not specify how long].

(3) A series of other stimulus-images is given, each one for 0.5s followed by a 1s delay, after which came the next stimulus. The number of such intervening stimuli ranged from 1-4 and was randomized each time. [The 1s delay is the ‘delay-period’ referred to in the results section. See Data analysis section p.5155B.]

(4) The final image of the series was the original stimulus image. In order to receive juice, the monkey needed to release the bar within 0.9s of the stimulus’ onset.

Some trials had no repetition of intervening images, others repeated ones in an ABBA pattern of sorts.

The results show that “Many of the neurons were stimulus-selective in that they responded better to some stimuli than to others.” [p.5156]

Some neurons were most active during the fixation period. Many others were highly active during the delay period (during which the monkeys viewed a blank screen).

More than half of the PF neurons (56%, or 82/145) showed significantly higher activity during the delay intervals compared with the spontaneous firing rate. For these cells, the average baseline firing rate was 11.9 spikes/sec, and the average level of delay activity was 16.0 spikes/sec. On average, delay activity was a 42% increase over baseline firing rate (SE 5 3.3%, range 4.9–351.2%).
[p.5157, boldface mine]

Certain cells were more active than others during delays, depending on the stimuli. [5157-5159]

Earl K. Miller; Cynthia A. Erickson;and Robert Desimone. Neural Mechanisms of Visual Working Memory in Prefrontal Cortex of the Macaque. The Journal of Neuroscience, 15 August 1996, 16(16): 5154-5167.

http://www.jneurosci.org/content/16/16/5154.full.pdf

http://www.jneurosci.org/content/16/16/5154.abstract

Funahashi et al. (1993) ‘Prefrontal neuronal activity in rhesus monkeys performing a delayed anti-saccade task’, notes

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Shintaro Funahashi; Matthew V. Chafee; Patricia S. Goldman-Rakic

Prefrontal neuronal activity in rhesus monkeys performing a delayed anti-saccade task

Brief summary:

Research suggests that our neurons store information about the past and also prepare for future actions using that information. Information about the past and the future, although not being present to our senses, is processed in our brains in the present.

Notes:

The authors examined the activity of 108 neurons in the dorsolateral prefrontal cortex, concentrating specifically on the delay period activity of these neurons. Two monkeys were to perform the following trial steps:

(1) Fixation on center of screen for 0.75 seconds (s)

(2) A visual cue stimulus is presented to the left or right of the fixation point for .5s [monkey may not move eyes yet]

(3) a period of delay of 3s when only the fixation point shows, and no cue [monkey may still not move eyes yet]

(4) After the delay ends, the fixation target disappears, and the monkey has 0.5s to move either to the side of the screen of the cue or toward the opposite side [depending on how it is to be trained], and upon success, was rewarded with a drop of juice.

[above from p754 of Funahashi et al.]

The results suggests that neurons may exhibit heightened activity during delay periods. But whether or not their activity increases seems to depend on what their role is. It seems that certain neurons are responsible for retaining data of certain kinds, mostly to do with the side that the stimulus is on, regardless of the resulting task or action.

Figure 2 illustrates a principal sulcus neuron whose increased firing in the delay is linked to the location of the visual cue, not the direction of the saccade. First, the neuron responded 98 ms after the onset of the visual stimulus when it appeared on the right but not the left (Fig. 2a, right; b, left; c, right). Second, it showed increased activity on ODR trials during the 3-s delay period, again only after the rightward stimulus (Fig. 2a, right). Third, and most relevant to the memory code, on anti-saccade trials, activity was again elevated in the delay period following the rightward stimulus (Fig. 2b, left), even though the saccade at the end of the delay was directed to the left. Fourth, on AS-ODR error trials, when the monkey mistakenly directed its saccades toward the stimulus, neuronal activity in the delay again reflected the rightward location of the stimulus (Fig. 2c, right). Thus, the response of this neuron in the delay was keyed | to the direction of a preceding stimulus, and not to that of the impending saccade, whether that saccade was correct or incorrect, and whether it was toward or away from the original stimulus. A similar pattern of stimulus dependence was found in the delay period activity of 30 out of 51 (or 59%) of the prefrontal neurons studied. If the 44 histologically verified principal sulcus neurons are considered alone, this percentage rises to 68%. The preponderance of stimulus-dependent delay period activity in this area is strong evidence of a prefrontal specialization for ideational processing that does not rely on a motor code and is not mediated by motor signals.
[p. 754]

[above from p754 of Funahashi et al.]

So the results seen above in figure 2 show us that there are neurons responsible for stimulus-coding. They are called stimulus-coding neurons. [754] During delay periods, these neurons increase activity for stimuli on the right side of the vision, regardless of reactions desired or executed.

There are also a response-coding function.

Activity was greater in the delay before rightward saccades, irrespective of whether the stimulus signalling the saccade had appeared on the right (Fig. 3a, right) or the left (Fig. 3b, right)
[754-755]

[above from p754 of Funahashi et al.]

So these neurons are called response-coding neurons. They are more active during delay periods when the animal will need to later move their eyes, in this case, to the right. This suggests that there are neurons which anticipate future actions.

So, different neurons have different functions, which combine to produce the complex behavior of waiting during delay periods all while retaining the past information during that time.

Independent response-coding and stimulus-coding mechanisms in prefrontal cortex might thus combine to mediate complex cognitive functions.

[755]

And thus because there are neurons which retain information and also ones which withhold actions that will use that information, brains exhibit the ability to process information about the past and future.

Stimulus-coding and response-coding delay period activities have been recorded previously from neurons in the principal sulcus on delayed-response tasks involving manual rather than oculomotor responses. The agreement between those results and our own, obtained with tasks recruiting different motor systems, suggests that the concurrent representation of past stimuli and future actions is a general principle of prefrontal operation. This segregation may enable the flexible re-association of stimulus representations and motor plans on a moment to moment basis on tasks in which stimulus-response relationships are not fixed. Stimulus and response-coding prefrontal neurons may directly or indirectly contribute to activity changes observed in motor areas, such as the supplementary motor area, the motor cortex, and the putamen, where target-dependent and limbdependent preparatory activity have also been recorded during a delayed-response task.
[755]

Shintaro Funahashi; Matthew V. Chafee; Patricia S. Goldman-Rakic. Prefrontal neuronal activity in rhesus monkeys performing a delayed anti-saccade task. Nature. 1993;365(6448):752-756.

http://www.cns.nyu.edu/~wendy/class/2006sp/reading9/Funahashi1993_nature.pdf

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The NeuroScience of Memory

Sperling (1960) ‘The Information Available in Brief Visual Presentation’, notes

 

Sperling (1967) ‘Successive approximations to a model for short term memory’, notes

Levitt (1971) ‘Transformed up‐down methods in psychoacoustics’, notes


Fuster and Alexander. (1971) ‘Neuronal Activity Related to Short-Term Memory’, notes

Weichselgartner & Sperling (1985) ‘Continuous Measurement of Visible Persistence’, notes

Funahashi et al. (1989) ‘Mnemonic coding of visual space in the monkey's dorsolateral prefrontal cortex,’ notes

 

Funahashi et al. (1993) ‘Prefrontal neuronal activity in rhesus monkeys performing a delayed anti-saccade task’, notes

Miller et al. (1996) ‘Neural Mechanisms of Visual Working Memory in Prefrontal Cortex of the Macaque’, notes

Rainer et al. (1999) ‘Prospective Coding for Objects in Primate Prefrontal Cortex’, notes

Heywood & Zihl (1999) Case Study of L.M.’s Inability to Perceive Motion, in their book chapter “Motion Blindness”, summary notes

 

 

 

 

Neurocomputation & Neural-network Computation

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Brain-Machine Interface Research

Some Recent Scientific Developments in Brain Machine Interface (for Robotic Prosthesis), Neuroplasticity, Neurocomputation, and Whole Brain Emulation

 

 

 

Fuster and Alexander. (1971) ‘Neuronal Activity Related to Short-Term Memory’, notes

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Fuster and Alexander

Neuronal Activity Related to Short-Term Memory

Brief summary:

This research suggests that our brains retain visual information, as sensory memory, by keeping parts of the brain active even after the stimulus is removed from sight and reaction to it is delayed.

Notes:

Monkeys look at 2 identical wooden blocks set to the right and left. The blocks are in an enclosure but there are doors under the blocks to allow the monkey to grab them [or things placed on them, or things placed in a well below things placed on the block]. The monkey’s vision is first obstructed, then the blind is lifted, allowing the monkey to see a piece of apple [available through one of the doors]. Then the blind is put back, which ends the ‘cue’ phase. Then there is a delay period, after which the doors are unlocked and the blind is lifted again. If the animal choses the right object [placed on the block, near which the apple piece once was], then it receives a reward. The monkeys were trained for 15 second delays. Then electrodes were inserted into their brains to measure activity.

What they found was that

Almost all the units investigated (57 in MD [nucleaus medialis dorsalis of the thalamus], 110 in prefrontal cortex) showed rather irregular patterns of spontaneous firing while the animal was at rest during inter-trial periods. The MD units generally displayed higher frequencies of firing than the cortical units did and, similar to units in other thalamic nuclei, a tendency to discharge in periodic groups or bursts of action potentials. [p653 BC]

In the course of delayed response trials the majority of units (58 percent of those in MD, 65 percent in prefrontal cortex) increased their spike activity to levels higher than those prevalent in intertrial periods. Some units exhibited a higher discharge rate during cue presentation, others during the delay, and still others during both cue and delay periods (Figs.1 and 2)

[From p653 of this article. As we can see in figure 1, there was more activity in the delay period than in the 20 second spontaneous firing segments. This is shown even more clearly in figure 3 below, from page 653]

The magnitude of the activation varied widely between different units, some reaching discharge levels more than tenfold higher than the spontaneous discharge level. Increased firing was in some units preceded by an inhibitory phase covering the beginning or the entirety of the cue presentation period. This inhibition was most conspicuous in units showing maximum dishcarge during the delay (Fig. 3). In some delay-activated units the increased firing | persisted throughout the delay period, slowly and irregularly declining toward baseline in the course of it. Figure 3 shows a unit activated for almost the entire duratino of delays longer than 1 minute.
[pp.653-654]

It is during the transition from cue to delay that apparently the greatest number of prefrontal units discharge at firing levels higher than the inter-trial baseline. […]

We believe that the excitatory reactions of neurons in MD and granular frontal cortex during delayed response trials are specifically related to the focusing of attention by the animal on information that is being or has been placed in temporary memory storage for prospective utilization.

[p654, boldfaces mine]

Fuster, J.M., G.E. Alexander. Neuron Activity Related to Short-Term memory. Science, New Series, Vol. 173 No.3997 (Aug. 13, 1971), pp.652-654. 
http://www.sciencemag.org/content/173/3997/652.abstract
http://www.cns.nyu.edu/~wendy/class/2006sp/reading9/Fuster_1971.pdf

Funahashi et al. (1989) ‘Mnemonic coding of visual space in the monkey's dorsolateral prefrontal cortex,’ notes

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Funahashi S, Bruce CJ, Goldman-Rakic PS.

“Mnemonic coding of visual space in the monkey's dorsolateral prefrontal cortex”

Brief summary:

This experiment suggests that we visually remember something because neurons used in our vision continue their activity even after the stimulus goes away.

 

Notes:

The authors are testing the activity of neurons used for vision during periods when the test subject needs to ‘remember’ what they saw, in absence of that stimulus. In their introduction they write:

Studies of single neuron activity in monkeys during performance of delayed-response tasks have also provided strong evidence of a prefrontal contribution to memory (4, 5, 15, 16,20,22, 37,4 1, 42, 46, 50-53). Although many types of neural activity have been found in the prefrontal cortex, neurons that show sustained activity during the delay period are particularly relevant to the issue of mnemonic processing. Usually these neurons increase their discharge rates following the brief cue presentation and continue firing tonically during the delay period until the response is executed (5, 15, 20, 22, 52, 53).
[p.331Bc]

The authors recorded activity from single neurons of monkeys. The monkeys were shown visual stimuli, and their eye moments were tracked. There is a fixation target, a small white spot in the middle of the screen. And there were peripheral visual cues, small filled white squares. The procedure went like this:

(1) 5s interval

(2) the fixation target appears at the center of the screen

(3) the monkey needed to be fixated for 0.75s (the fixation period)

(4) after those 0.75s, the peripheral visual cue was presented for 0.5s (the cue period) at one of many peripheral locations. The monkey was only rewarded if they maintained fixation in the center during the cue period and also during a subsequent delay period.

(5) delay period (monkey must stay fixated at center) of either 1.5, 3 or 6 (normally 3) seconds long.

(6) response period, within the next 0.5s after the delay, monkey must make saccadic eye movement to the cue’s location (to get reward). [332AB]

Our concern here lies with the neuronal activity. The measured the ‘discharge rate’ (p.334Aa) of 319 neurons in the prefrontal cortex (336Ba)

[Image from Funahashi et al., p.332.]

[The above diagrams show that during the delay period, when the monkey needs to visually ‘remember’ the cue’s location, there is heightened neuron activity.]

Findings: They find that of the 288 neurons (in the prefrontal cortex, within and surrounding the principal sulcus (PS)) “170 exhibited task-related activity during at least one phase of this task and, of these, 87 showed significant excitation or inhibition of activity during the delay period relative to activity during the intertrial interval,” and “For 50 of the 87 PS neurons, activity during the delay period was significantly elevated above the neuron’s spontaneous rate for at least one cue location.”

[331Bd]

S. Funahashi , C. J. Bruce , P. S. Goldman-Rakic. Mnemonic coding of visual space in the monkey's dorsolateral prefrontal cortex. Journal of NeurophysiologyPublished 1 February 1989Vol. 61no. 331-349.
http://www.ncbi.nlm.nih.gov/pubmed/2918358

 

Katz and Sherry’s [Pt.4.6] “Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 4.6 ‘Assignable Versus Unassignable’, summary

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Mikhail G. Katz  and David Sherry

“Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”

4. Cum Prodiisset

 

4.6 Assignable Versus Unassignable

Brief Summary:

In Cum Prodiisset Leibniz works with assignable (finite) and unassignable (infinitesimal / infinite) values. On account of his law of continuity, we can use the same mathematical operations even when we substitute one for the others.

Summary

 

Leibniz’ infinitesimal values dx and dy are ‘unassignable,’ and Leibniz writes that

the unassignables dx and dy may be substituted for them by a method of supposition even in the case when they are evanescent (Child [Leibniz] 1920, p. 153).
[KS 583]

KS will examine Leibniz’ multiplicative example:

ay = xv

[We will look briefly at Leibniz’ procedures to follow Katz and Sherry. Leibniz writes: “Let ay = xv, then a(d)y = x(d)v + v(d)x.” Leibniz then adds to their corresponding values dy, dx, and dv:

Proof. ay + ady = (x + dx) (v+ dv) “

then expanding

= xv + xdv + vdx +dxdv.

And so together:

ay + ady = (x + dx) (v+ dv) = xv + xdv + vdx +dxdv

or

ay + ady = xv + xdv + vdx +dxdv

(note: there is a transcription error in the translation. In the following, the original has ‘xv’ but the translation has instead ‘xy’.)

We then remove ay and xy (perhaps because they are repeated now in their infinitesimal renditions)

ady = xdv + vdx + dxdv

(then dividing both sides by dx)

This is the formulation KS move next to.]

“Simplifying the differential quotient, Leibniz obtains”

[It seems then that Leibniz will replace the infinitesimals dy and dx with finite differences (d)y and d(x), but he will not do the same for the second d(v) for some reason, leaving it as infinitesimal. Returning to KS:]

At this point Leibniz proposes to transfer ‘‘the matter, as we may, to straight lines that never become evanescent’’, obtaining

KS explain that (d)y and (d)x are assignable, while dv is ‘superfluous’ as ‘it alone can be evanescent’ [KS 583, quoting Leibniz].

[This means that dv can be treated as zero and removed from the equation. Then, we multiple both sides by (d)x to obtain:

Then Leibniz divides both sides by a and by (d)x to obtain:

Then Leibniz notes that (d)y / (d)x “always” equals dy/dx. Thus we can substitute them. This is an application of the principle of continuity again:

Also, since (dy) : (d)x always = dy : dx, it will be allowable to suppose this is true in the case when dy, dx become evanescent, and to say that dy : dx = x + v : a, or ady = xdv + vdx.
(Leibniz 154)

(above by multiplying both sides by a and dx).]

But the law of homogeneity is not mentioned here in Cum Prodiisset, so we do not here have sufficient rational for that step in the operation.

The authors then show a case in the Leibniz text of division by second differentials, which they say is incompatible with the nilsquare approach. [The author’s mention the nilsquare approach in section 4.1]

Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

Regarding the Leibniz text:

English references from:

Leibniz. The Early Mathematical Manuscripts of Leibniz. Trans. J.M. Child. Mineola, NY: Dover, 2005 [1920 Open Court].
1920 Edition available at archive.org:
https://archive.org/details/earlymathematic01gerhgoog

Latin references from:

Leibniz. Historia et origo calculi differentialis. Ed. C.I. Gerhardt. Hannover: Im Verlage der Hahn'schen Hofbuchhandlung, 1846]
Available at archive.org:
https://archive.org/details/historiaetorigo00gerhgoog

Katz and Sherry’s [Pt.4.5] “Leibniz’ Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 4.5 ‘Mathematical Implementation of Status Transitus,’ summary

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Mikhail G. Katz  and David Sherry

“Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”

4. Cum Prodiisset

 

4.5 Mathematical Implementation of Status Transitus

Brief Summary:

In Cum Prodiisset Leibniz discusses his Law of Continuity, and he provides some examples. Katz and Sherry will examine some of the mathematical procedures involved in these examples. What we find is that the same formulations and procedures are used regardless of whether we are dealing with finite or infinite values. This is because the law of continuity postulates that what holds for the finite holds for the infinite (and infinitesimal), [because there is a continuous transition from one to the other.]

Summary

 

In section 4.2 and 4.4, Katz and Sherry (KS) discuss three applications of Leibniz’ law of continuity and infinitesimals in his Cum Prodiisset. Now they will examine them more mathematically.

Leibniz distinguishes assignable finite quantity d(x) from infinitesimal value dx.

The assignable quantity (d)x passes via infinitesimal dx on its way to absolute 0. Then the infinitesimal dx is the terminus, or the status transitus. Zero is merely the shadow of the infinitesimal. This particular status transitus is the foundation rock of the Leibnizian definition of the differential quotient.
[KS 581]

[Recall Leibniz’ example of finding the tangent to a parabola. He writes:

“let ₁X₂X, the difference between A₁X and A₂X, be called dx;”

“and similarly, let D₂Y, the difference between ₁X₁Y and ₂X₂Y, be called dy.” (Leibniz 151)

Then, we shrink dx and dy down to find the tangent.

(Animated diagram by Corry Shores, using OpenOffice Draw and Unfreez)

Here Leibniz does not notate a difference between dx and d(x), but we would think of the finitely large line as being d(x) and the infinitesimally small one as dx.

KS also refer to Z as being the shadow of the infinitesimal. In the appendix they explain the shadow as being a real number assigned to hyperreal infinitesimals. It is the next closest value, only infinitesimally different. In these cases below, the shadow is like rounding the infinitesimal value to the nearest real number.]]

Let’s recall the first example Katz and Sherry examine:

(1) In the context of a discussion of parallel lines, he writes: when the straight line BP ultimately becomes parallel to the straight line VA, even then it converges toward it or makes an angle with it, only that the angle is then infinitely small (Child 1920, p. 148).
[[KS579]]

[Image from Leibniz/Child 148]

[We animated it thus:

(Animated diagram by Corry Shores, using OpenOffice Draw and Unfreez)

]

In KS’s rendition, they have us think of this diagram on its side with the moving line intercepting the x axis. In their formulation, H is the x-intercept value.

Example (1) of parallel lines can be elaborated as follows. Let’s follow Leibniz in building the line through the point (0,1) parallel to the x-axis in the plane. Line L_H with y-intercept 1 and x-intercept H is given by y = x / H .
[KS 581]

[Referring back to Leibniz’ diagram, the y-intercept here is like point P, which remains the same, all while the x-intercept, equivalent to point C, moves to greater distances.]

We will move the x-intercept to infinity. [KS later refer to the ‘finite portion of L_H.’ Perhaps one way to picture this is to think that the lines meet way off into infinity, but they ‘drag’ the finite part of the line all the way up such that it is only infinitesimally away from being y = 1. Yet, somewhere the lines must meet, which means it cannot stay at y = 1 everywhere. So does that mean that in the infinite part, it is no longer infinitely close to being parallel? I am not sure, but I think it is always infinitely close to being parallel, but just not always being infinitely close to y = 1. So far off into infinity, it will maybe be infinitely close to being parallel to y = 0.5, and so on diminishing. For every y position of the line in its infinite part, there would be an infinite stretch both ways, making every position be nearly parallel to the line.]

Now let H be infinite. The resulting line LH has negative infinitesimal slope, meets the x-axis at an infinite point, and forms an infinitesimal angle with the x-axis at the point where they meet. We will denote by st(x) the assignable (i.e., real) shadow of a finite x. [KS 581]

[They now will formulate for finite values for the x and y locations included in Line H.]

Then every finite point (x,y) ∈ L_H satisfies

[by substituting the formula for y. Now, because we are supposing H to be infinite, that means you have a finite value divided by an infinite one, which means it is nearly zero, which means that the y value is very close to 1, and thus:]

Hence the finite portion of L_H is infinitely close to the line y = 1. The line y = 1 is parallel to the x-axis, and is merely the shadow of the inassignable L_H. Thus, the parallel line is constructed by varying the oblique line depending on a parameter. Such variation comprises the status transitus L_H defined by an infinite value of H.
[KS 582]

Now recall example 3:

(3) Finally, a conception of a parabola expressed by means of an ellipse with an infinitely removed focal point is articulated in the following terms: a parabola is the ultimate form of an ellipse, in which the second focus is at an infinite distance from the given focus nearest to the given vertex (Child 1920, p. 148).
[KS 579]

[[Recall from our prior discussion that by moving one focus of the ellipse infinitely away from the other, the ellipse transforms into a parabola.

(Animated diagram by Corry Shores, using OpenOffice Draw and Unfreez)

]

KS will provide a formulation to illustrate this better.

To implement example (3), let’s follow Leibniz in deforming an ellipse, via a status transitus, into a parabola. The ellipse with vertex (apex) at (0,-1) and with foci at the origin and at (0,H) is given by

[They then square both sides and expand the values:]

Then they kept the radical on the left side, moving the rest over:

Then they square both sides, and reduce to get:

[We notice that even though H is an infinite number, they still apply operations on the figures that normally apply to finite numbers. But whether or not that is mathematically admissible is uncertain, hence it the ‘postulate’ of the law of continuity that we can treat them in like manner.]

The calculation (4.1) through (4.4) depends on the following habits of general reasoning (to echo Child’s translation) with assignable quantities, which are generalized to apply to inassignable quantities (such as the terminus/status transitus) in accordance with the law of continuity:

• squaring undoes a radical;

• the binomial formula;

• terms in an equation can be transferred to the other side; etc.

General reasoning of this type is familiar in the realm of ordinary finite real numbers, but why does it remain valid when applied to the realm of infinite or infinitesimal numbers? The validity of transfering such general reasoning originally instituted in the finite realm, to the realm of the infinite is postulated by Leibniz’s law of continuity.
[KS 582]

[Because the same general reasoning is used for finite and infinite, that might give a ‘between’ status or ambiguous status to the status transitus. Notice that we are still using a formulation for an ellipse. And yet, the resulting figure is in transit to being a parabola. Off in infinity, it might be still elliptical. But it draws out the finite part such that only parabolic parts remain within the realm of assignable values.]

We therefore apply Leibniz’s law of continuity to Eq. (4.4) for an infinite H. The resulting entity is still an ellipse of sorts, to the extent that it satisfies all of the Eqs. (4.1) to (4.4). However, this entity is no longer finite. It represents a Leibnizian status transitus between ellipse and parabola. This status transitus has foci at the | origin and at an infinitely distant point (0, H).
[KS 582|583, boldface mine]

[[Recall that we left off before with:

It seems in the following that KS will assign real numbers so to find the “real shadow” of this figure, that is, the finite assignable values it is infinitely close to.]]

Assuming x and y are finite, we set x₀ = st(x) and y₀ = st(y), to obtain a real shadow of this entity:

then simplifying:

Thus, the finite portion of the status transitus (4.4) is infinitely close to its shadow (4.5), namely the real parabola y ¼ x2 4 1 (in Leibniz’s terminology as translated by Child, ‘‘it is really true’’ that this parabola has no focus at infinity—see Sect. 4.4). This is the kind of payoff Leibniz is seeking with his law of continuity.
[KS 583]

Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

Examples of Status Transitus / Status Terminus and the Law of Continuity in Leibniz’ Cum Prodiisset

summary by Corry Shores
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Leibniz

 

Examples of Status Transitus / Status Terminus and the Law of Continuity in

 Cum Prodiisset

 

Brief Summary:

In Cum Prodiisset Leibniz discusses his Law of Continuity. According to one formulation, in a continuous transition, the final ending (the terminus) of the transition may be included with that transition. He provides a number of examples to illustrate. They show changes from opposing states happening continuously and by means of an infinitesimally small variation, for example, unequal and equal, motion and rest, convergent and parallel, and enclosed and open. On account of the law of continuity, the same calculatory procedures apply both when the values are finite and also as they pass into the infinite.

Summary

 

Before giving his examples of status transitus [/status terminus] (a concept bound up with his law of continuity), he first formulates it thus:

In any supposed transition, ending in any terminus, it is permissible to institute a general reasoning, in which the final terminus may also be included. [Leibniz 147]

 

[[See especially Katz and Sherry’s discussion of this text, section 4 of “Leibniz’ Infinitesimals”]] Leibniz will repeat this idea of ‘a general reasoning’ for his examples. One of them involves calculations, so we will skip first to that one to gain a more technical understanding of the term. [The following, up to ‘Example 1’ is quotation except for material in double brackets. The main idea we will obtain is that a general reasoning allows the same calculating procedures to continue even after values shift from finite to infinitesimal.]

“

First of all, the sense in which the phrase "dy is the element of y," is to be taken will best be understood by considering a line A  referred to a straight line AX as axis.

Let the curve AY be a parabola,

and let the tangent at the vertex A be taken as the axis. [[Note, the Latin original indicates the axis as AX: et assumtos Axis AX sit tangens parabolae in vertice A. (p.44)]]

If AX is called x,

and AY, y,

[[given the way that dy is defined below, y could instead be the distance from A to Y, when Y is understood as a vertical axis.

]]

and the latus-rectum is a [[the latus-rectum is often defined as the line going through the focus, perpendicular to the axis, with endpoints on the parabola. It does not seem to be shown here]], the equation to the parabola will be xx = ay, and this holds good at every point. Now, let A₁X= x,

and ₁X₁Y = y

and from the point ₁Y let fall a perpendicular ₁YD to some greater ordinate ₂X₂Y that follows,

and let ₁X₂X, the difference between A₁X and A₂X, be called dx;

and similarly, let D₂Y, the difference between ₁X₁Y and ₂X₂Y, be called dy.

Then, since y = xx: a, by the same law,

[[

xx = ay
xx / a = ay / a
xx /a = y
y = xx / a

]]

we have

y+ dy = xx + 2xdx + dxdx, : a ;

[[regarding the above: since we are adding dy to y, perhaps then the other side needs to be (x + dx)(x + dx), because we would be adding dx to both x’s just as we add dy to all cases of y. Then we would obtain the above formulation by expanding it.]]

and taking away the y from the one side and the xx: a from the other, we have left

dy: dx = 2x + dx : a ;

[[It seems we can remove the y = xx / a, perhaps because the same formulation repeats with the dy and dx. That leaves us with:

dy = (2xdx + dxdx) / a ;

Then we move a dx to the other side:

dy / dx = (2xdx + dxdx) / dx(a)

Producing the formula again:

dy / dx = 2x + dx / a

]]

and this is a general rule, expressing the ratio of the difference of the ordinates to the difference of the abscissae, [[the formulation y = xx / a gave us the relation of the ordinates to the abscissae. Then we reformulated it to give us the relation between the differences of the abscissae and ordinate by means of dy and dx.]]

or, if the chord ₁Y₂Y is produced until it meets the axis in T,

then the ratio of the ordinate ₁X₁Y to T₁X, the part of the axis intercepted between the point of intersection and the ordinate, will be as 2x+ dx to a. Now, since by our postulate it is permissible to include under the one general reasoning the case also in which the ordinate ₂X ₂Y is moved up nearer and nearer to the fixed ordinate ₁X ₁Y until it ultimately coincides with it, it is evident that in this case dx becomes equal to zero and should be neglected, and thus it is clear that, since in this case T₁Y is the tangent, ₁X₁Y is to T₁X as 2x is to a.

[[Leibniz then seems to state that what he means with the law of continuity is that we can regard as equivalent cases both with evanescent values and with them removed.]]

Hence, it may be seen that there is no need in the whole of our differential calculus to say that those things are equal which have a difference that is infinitely small, but that those things can be taken as equal that have not any difference at all, provided that the calculation is supposed to be general, including both the cases | in which there is a difference and in which the difference is zero; and provided that the difference is not assumed to be zero until the calculation is purged as far as is possible by legitimate omissions, and reduced to ratios of non-evanescent quantities, and we finally come to the point where we apply our result to the ultimate case.

[[Thus to ‘include under one general reasoning seems to mean in this case that we consider the formula for the parabola as holding for all values of ₁Y₂Y, including when that value vanishes to zero. Perhaps then in the examples that precede this one in the text, which we will discuss below, ‘under one general reasoning’ might as well mean that whatever formula we regard as applying in the finite cases as well apply in the infinitesimal case.]]

” [The above quoted material from Leibniz 151-152]

Example 1:

We consider quantities A and B, where B is larger than A. However, A is diminishing until equaling B. Nonetheless, we can include its value at B as belonging in unbroken continuum with the prior values, even though their states are contrary [[when vanishing to B, A is both equal and unequal to B]]. Here is the passage as quotation:

if A and B are any two quantities, of which the former is the greater and the latter is the less, and while B remains the same, it is supposed that A is continually diminished, until A becomes equal to B ; then it will be permissible to include under a general reasoning the prior cases in which A was greater than B, and also the ultimate case in which the difference vanishes and A is equal to B. [Leibniz 147]

Example 2:

Consider two bodies in motion, A and B. A’s velocity is continuously diminishing to zero (rest) all while B’s remains the same. Even though A’s motion is coming to a rest, we can include all its variations in speed, including its coming-to-rest state, with B’s motion. [[See quoted text below. Perhaps we are to think of A and B as beginning at the same speed, so to both illustrate the continuity of change with the dichotomy of states, motion and rest]]

Similarly, if two bodies are in motion at the same time, and it is assumed that while the motion of B remains the same, the velocity of A is continually diminished until it vanishes altogether, or the speed of A becomes zero ; it will be permissible to include this case with the case of the motion of B under one general reasoning. [Leibniz 147]

Example 3:

Consider of we have a line converging with another at some angle. We pivot the line on some fixed point (P above), extending the line so that it continues to converge with the other line. The angle continuously diminishes. When it reaches the infinitely small, the lines are becoming parallel. Even though the line’s position is discontinuous, this transition is included with all prior ones.

[Animated diagram by Corry Shores, using OpenOffice Draw and Unfreez]

From the translation:

We do the same thing in geometry, when two straight lines are taken, produced in any manner, one VA being given in position or remaining in the same site, the other BP passing through a given point P, and varying in position while the point P remains fixed ; at first indeed converging toward the line VA and meeting it in the point C ; then, as the angle of inclination VCA is continually diminished, meeting VA in some more remote point (C), until at length from BP, through the position (B)P, it comes | to βP, in which the straight line no longer converges toward VA, but is parallel to it, and C is an impossible or imaginary point. With this supposition it is permissible to include under some one general reasoning not only all the intermediate cases such as (B)P but also the ultimate case βP. [Leibniz trans 147. Above case of  “as the angle of inclination VCA is continually diminished” should have BCA instead, “deinde si angulus inclinationis, ut BCA continue minuatur” ]

Example 4:

Consider an ellipse. One focus moves away from other. In the infinitesimally small movement from finite to infinitely far, the figure changes from ellipse to parabola, and thus from enclosed to open.

[Animated diagram by Corry Shores, using OpenOffice Draw and Unfreez]

From the text:

Hence also it comes to pass that we include as one case ellipses and the parabola, just as if A is considered to be one focus of an ellipse (of which V is the given vertex), and this focus remains fixed, while the other focus is variable as we pass from ellipse to ellipse, until at length (in the case when the line BP, by its intersection with the line VA, gives the variable focus) the focus C becomes evanescent73 or impossible, in which case the ellipse passes into a parabola. Hence it is permissible with our postulate that a parabola should be considered with ellipses under a common reasoning. [ft 73: “ The term is here used with the idea of "vanishing into the far distance." ”]Just as it is common practice to make use of this method in geometrical constructions, when they include under one general construction many different cases, noting that in a certain case the converging straight line passes into a parallel straight line, the angle between it and another straight line vanishing. [Leibniz 148]

Leibniz continues, refering to these examples:

Moreover, from this postulate arise certain expressions which are generally used for the sake of convenience, but seem to contain an absurdity, although it is one that causes no hindrance, when its proper meaning is substituted. For instance, we speak of an imaginary point of intersection as if it were a real point, in the same manner as · in algebra imaginary roots are considered as accepted numbers. Hence, preserving the analogy, we say that, when the straight line BP ultimately becomes parallel to the straight line VA, even then it converges toward it or makes an angle with it, only that the angle is then infinitely small; similarly, when a body ultimately comes to rest, it is still said to have a velocity, but one that is infinitely small ; and, when one straight line is equal to another, it is said to be unequal to it, but that the difference is infinitely small ; and that a parabola is the ultimate form of an ellipse, in which the second focus is at an infinite distance from the given focus nearest to the given vertex, or in which the ratio of PA to AC, or the angle BCA, is infinitely small. [Leibniz 148]

Bibliography:

English references from:

Leibniz. The Early Mathematical Manuscripts of Leibniz. Trans. J.M. Child. Mineola, NY: Dover, 2005 [1920 Open Court].

1920 Edition available at archive.org:

https://archive.org/details/earlymathematic01gerhgoog

Latin references from:

Leibniz. Historia et origo calculi differentialis. Ed. C.I. Gerhardt. Hannover: Im Verlage der Hahn'schen Hofbuchhandlung, 1846]

Available at archive.org:

https://archive.org/details/historiaetorigo00gerhgoog

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html