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I had hoped that the white moderate would understand that the present tension in the South is a necessary phase of the transition from an obnoxious negative peace, in which the Negro passively accepted his unjust plight, to a substantive and positive peace, in which all men will respect the dignity and worth of human personality. Actually, we who engage in nonviolent direct action are not the creators of tension.
We merely bring to the surface the hidden tension that is already alive. We bring it out in the open, where it can be seen and dealt with. Like a boil that can never be cured so long as it is covered up but must be opened with all its ugliness to the natural medicines of air and light, injustice must be exposed, with all the tension its exposure creates, to the light of human conscience and the air of national opinion before it can be cured.
In your statement you assert that our actions, even though peaceful, must be condemned because they precipitate violence. But is this a logical assertion? Isn't this like condemning a robbed man because his possession of money precipitated the evil act of robbery? Isn't this like condemning Socrates because his unswerving commitment to truth and his philosophical inquiries precipitated the act by the misguided populace in which they made him drink hemlock? Isn't this like condemning Jesus because his unique God consciousness and never ceasing devotion to God's will precipitated the evil act of crucifixion?
We must come to see that, as the federal courts have consistently affirmed, it is wrong to urge an individual to cease his efforts to gain his basic constitutional rights because the quest may precipitate violence. Society must protect the robbed and punish the robber. I had also hoped that the white moderate would reject the myth concerning time in relation to the struggle for freedom. I have just received a letter from a white brother in Texas. He writes: "All Christians know that the colored people will receive equal rights eventually, but it is possible that you are in too great a religious hurry.
It has taken Christianity almost two thousand years to accomplish what it has. The teachings of Christ take time to come to earth. Actually, time itself is neutral; it can be used either destructively or constructively. More and more I feel that the people of ill will have used time much more effectively than have the people of good will. We will have to repent in this generation not merely for the hateful words and actions of the bad people but for the appalling silence of the good people.
Human progress never rolls in on wheels of inevitability; it comes through the tireless efforts of men willing to be co workers with God, and without this hard work, time itself becomes an ally of the forces of social stagnation. We must use time creatively, in the knowledge that the time is always ripe to do right. Now is the time to make real the promise of democracy and transform our pending national elegy into a creative psalm of brotherhood.
Now is the time to lift our national policy from the quicksand of racial injustice to the solid rock of human dignity. You speak of our activity in Birmingham as extreme. At first I was rather disappointed that fellow clergymen would see my nonviolent efforts as those of an extremist. I began thinking about the fact that I stand in the middle of two opposing forces in the Negro community. One is a force of complacency, made up in part of Negroes who, as a result of long years of oppression, are so drained of self respect and a sense of "somebodiness" that they have adjusted to segregation; and in part of a few middle-class Negroes who, because of a degree of academic and economic security and because in some ways they profit by segregation, have become insensitive to the problems of the masses.
The other force is one of bitterness and hatred, and it comes perilously close to advocating violence. It is expressed in the various black nationalist groups that are springing up across the nation, the largest and best known being Elijah Muhammad's Muslim movement. Nourished by the Negro's frustration over the continued existence of racial discrimination, this movement is made up of people who have lost faith in America, who have absolutely repudiated Christianity, and who have concluded that the white man is an incorrigible "devil.
I have tried to stand between these two forces, saying that we need emulate neither the "do nothingism" of the complacent nor the hatred and despair of the black nationalist. For there is the more excellent way of love and nonviolent protest. I am grateful to God that, through the influence of the Negro church, the way of nonviolence became an integral part of our struggle.
If this philosophy had not emerged, by now many streets of the South would, I am convinced, be flowing with blood. And I am further convinced that if our white brothers dismiss as "rabble rousers" and "outside agitators" those of us who employ nonviolent direct action, and if they refuse to support our nonviolent efforts, millions of Negroes will, out of frustration and despair, seek solace and security in black nationalist ideologies--a development that would inevitably lead to a frightening racial nightmare.
Oppressed people cannot remain oppressed forever.
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The yearning for freedom eventually manifests itself, and that is what has happened to the American Negro. Something within has reminded him of his birthright of freedom, and something without has reminded him that it can be gained. Consciously or unconsciously, he has been caught up by the Zeitgeist, and with his black brothers of Africa and his brown and yellow brothers of Asia, South America and the Caribbean, the United States Negro is moving with a sense of great urgency toward the promised land of racial justice.
If one recognizes this vital urge that has engulfed the Negro community, one should readily understand why public demonstrations are taking place. The Negro has many pent up resentments and latent frustrations, and he must release them. So let him march; let him make prayer pilgrimages to the city hall; let him go on freedom rides -and try to understand why he must do so. If his repressed emotions are not released in nonviolent ways, they will seek expression through violence; this is not a threat but a fact of history. So I have not said to my people: "Get rid of your discontent. And now this approach is being termed extremist.
But though I was initially disappointed at being categorized as an extremist, as I continued to think about the matter I gradually gained a measure of satisfaction from the label. Was not Jesus an extremist for love: "Love your enemies, bless them that curse you, do good to them that hate you, and pray for them which despitefully use you, and persecute you. Will we be extremists for hate or for love? Will we be extremists for the preservation of injustice or for the extension of justice?
In that dramatic scene on Calvary's hill three men were crucified. We must never forget that all three were crucified for the same crime--the crime of extremism. Two were extremists for immorality, and thus fell below their environment. The other, Jesus Christ, was an extremist for love, truth and goodness, and thereby rose above his environment. Perhaps the South, the nation and the world are in dire need of creative extremists. I had hoped that the white moderate would see this need. Perhaps I was too optimistic; perhaps I expected too much.
I suppose I should have realized that few members of the oppressor race can understand the deep groans and passionate yearnings of the oppressed race, and still fewer have the vision to see that injustice must be rooted out by strong, persistent and determined action.
I am thankful, however, that some of our white brothers in the South have grasped the meaning of this social revolution and committed themselves to it. They are still all too few in quantity, but they are big in quality. Others have marched with us down nameless streets of the South. They have languished in filthy, roach infested jails, suffering the abuse and brutality of policemen who view them as "dirty nigger-lovers.
Let me take note of my other major disappointment. I have been so greatly disappointed with the white church and its leadership. Of course, there are some notable exceptions. I am not unmindful of the fact that each of you has taken some significant stands on this issue. I commend you, Reverend Stallings, for your Christian stand on this past Sunday, in welcoming Negroes to your worship service on a nonsegregated basis. I commend the Catholic leaders of this state for integrating Spring Hill College several years ago.
But despite these notable exceptions, I must honestly reiterate that I have been disappointed with the church. I do not say this as one of those negative critics who can always find something wrong with the church. I say this as a minister of the gospel, who loves the church; who was nurtured in its bosom; who has been sustained by its spiritual blessings and who will remain true to it as long as the cord of life shall lengthen.
When I was suddenly catapulted into the leadership of the bus protest in Montgomery, Alabama, a few years ago, I felt we would be supported by the white church. I felt that the white ministers, priests and rabbis of the South would be among our strongest allies. Instead, some have been outright opponents, refusing to understand the freedom movement and misrepresenting its leaders; all too many others have been more cautious than courageous and have remained silent behind the anesthetizing security of stained glass windows. In spite of my shattered dreams, I came to Birmingham with the hope that the white religious leadership of this community would see the justice of our cause and, with deep moral concern, would serve as the channel through which our just grievances could reach the power structure.
I had hoped that each of you would understand. But again I have been disappointed. I have heard numerous southern religious leaders admonish their worshipers to comply with a desegregation decision because it is the law, but I have longed to hear white ministers declare: "Follow this decree because integration is morally right and because the Negro is your brother. In the midst of a mighty struggle to rid our nation of racial and economic injustice, I have heard many ministers say: "Those are social issues, with which the gospel has no real concern.
I have traveled the length and breadth of Alabama, Mississippi and all the other southern states. On sweltering summer days and crisp autumn mornings I have looked at the South's beautiful churches with their lofty spires pointing heavenward. I have beheld the impressive outlines of her massive religious education buildings. Over and over I have found myself asking: "What kind of people worship here?
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Where were their voices when the lips of Governor Barnett dripped with words of interposition and nullification? Where were they when Governor Wallace gave a clarion call for defiance and hatred? Where were their voices of support when bruised and weary Negro men and women decided to rise from the dark dungeons of complacency to the bright hills of creative protest?
Yes, these questions are still in my mind. In deep disappointment I have wept over the laxity of the church. But be assured that my tears have been tears of love. There can be no deep disappointment where there is not deep love. Yes, I love the church. How could I do otherwise? I am in the rather unique position of being the son, the grandson and the great grandson of preachers.
Yes, I see the church as the body of Christ.
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But, oh! How we have blemished and scarred that body through social neglect and through fear of being nonconformists. There was a time when the church was very powerful--in the time when the early Christians rejoiced at being deemed worthy to suffer for what they believed. In those days the church was not merely a thermometer that recorded the ideas and principles of popular opinion; it was a thermostat that transformed the mores of society. Whenever the early Christians entered a town, the people in power became disturbed and immediately sought to convict the Christians for being "disturbers of the peace" and "outside agitators.
Small in number, they were big in commitment. They were too God-intoxicated to be "astronomically intimidated. Things are different now. So often the contemporary church is a weak, ineffectual voice with an uncertain sound. So often it is an archdefender of the status quo. Far from being disturbed by the presence of the church, the power structure of the average community is consoled by the church's silent--and often even vocal--sanction of things as they are. But the judgment of God is upon the church as never before. If today's church does not recapture the sacrificial spirit of the early church, it will lose its authenticity, forfeit the loyalty of millions, and be dismissed as an irrelevant social club with no meaning for the twentieth century.
Every day I meet young people whose disappointment with the church has turned into outright disgust. Perhaps I have once again been too optimistic. Is organized religion too inextricably bound to the status quo to save our nation and the world? Perhaps I must turn my faith to the inner spiritual church, the church within the church, as the true ekklesia and the hope of the world.
But again I am thankful to God that some noble souls from the ranks of organized religion have broken loose from the paralyzing chains of conformity and joined us as active partners in the struggle for freedom.
Making fingers and words count in a cognitive robot
They have left their secure congregations and walked the streets of Albany, Georgia, with us. They have gone down the highways of the South on tortuous rides for freedom. Yes, they have gone to jail with us. Some have been dismissed from their churches, have lost the support of their bishops and fellow ministers. But they have acted in the faith that right defeated is stronger than evil triumphant.
Their witness has been the spiritual salt that has preserved the true meaning of the gospel in these troubled times. They have carved a tunnel of hope through the dark mountain of disappointment. I hope the church as a whole will meet the challenge of this decisive hour. But even if the church does not come to the aid of justice, I have no despair about the future. I have no fear about the outcome of our struggle in Birmingham, even if our motives are at present misunderstood.
We will reach the goal of freedom in Birmingham and all over the nation, because the goal of America is freedom. Abused and scorned though we may be, our destiny is tied up with America's destiny. Before the pilgrims landed at Plymouth, we were here. Before the pen of Jefferson etched the majestic words of the Declaration of Independence across the pages of history, we were here. For more than two centuries our forebears labored in this country without wages; they made cotton king; they built the homes of their masters while suffering gross injustice and shameful humiliation -and yet out of a bottomless vitality they continued to thrive and develop.
If the inexpressible cruelties of slavery could not stop us, the opposition we now face will surely fail. We will win our freedom because the sacred heritage of our nation and the eternal will of God are embodied in our echoing demands. Before closing I feel impelled to mention one other point in your statement that has troubled me profoundly. You warmly commended the Birmingham police force for keeping "order" and "preventing violence. I doubt that you would so quickly commend the policemen if you were to observe their ugly and inhumane treatment of Negroes here in the city jail; if you were to watch them push and curse old Negro women and young Negro girls; if you were to see them slap and kick old Negro men and young boys; if you were to observe them, as they did on two occasions, refuse to give us food because we wanted to sing our grace together.
I cannot join you in your praise of the Birmingham police department. It is true that the police have exercised a degree of discipline in handling the demonstrators. In this sense they have conducted themselves rather "nonviolently" in public. But for what purpose? To preserve the evil system of segregation. Over the past few years I have consistently preached that nonviolence demands that the means we use must be as pure as the ends we seek.
I have tried to make clear that it is wrong to use immoral means to attain moral ends. But now I must affirm that it is just as wrong, or perhaps even more so, to use moral means to preserve immoral ends. Perhaps Mr. Connor and his policemen have been rather nonviolent in public, as was Chief Pritchett in Albany, Georgia, but they have used the moral means of nonviolence to maintain the immoral end of racial injustice.
Eliot has said: "The last temptation is the greatest treason: To do the right deed for the wrong reason. I wish you had commended the Negro sit inners and demonstrators of Birmingham for their sublime courage, their willingness to suffer and their amazing discipline in the midst of great provocation. One day the South will recognize its real heroes. They will be the James Merediths, with the noble sense of purpose that enables them to face jeering and hostile mobs, and with the agonizing loneliness that characterizes the life of the pioneer.
They will be old, oppressed, battered Negro women, symbolized in a seventy two year old woman in Montgomery, Alabama, who rose up with a sense of dignity and with her people decided not to ride segregated buses, and who responded with ungrammatical profundity to one who inquired about her weariness: "My feets is tired, but my soul is at rest. One day the South will know that when these disinherited children of God sat down at lunch counters, they were in reality standing up for what is best in the American dream and for the most sacred values in our Judaeo Christian heritage, thereby bringing our nation back to those great wells of democracy which were dug deep by the founding fathers in their formulation of the Constitution and the Declaration of Independence.
Never before have I written so long a letter. The iCub is not provided with ears, so the auditory input i. From the waveform, we extracted the mel-frequency cepstral coefficients MFCC to represent each number word from one to ten, using the Slaney's auditory toolbox 2. MFCC technique combines an auditory filter-bank with a cosine transform to give a rate representation roughly similar to the auditory system Davis and Mermelstein, After supervised training, it is able to represent the correspondence between numbers from 1 to 10 and the internal representations i.
The role of the competitive layer classifier is to simulate the final processing of the numbers, after a number is correctly classified into its class, the appropriate action can be started, e. Schematic of the Robot's Cognitive System. Bold links indicate a full one-to-all connection between each layer, while dotted links are direct one-to-one connections.
Note that the system's external inputs coincide with the outputs, indeed proprioceptive information from motor and auditory systems is an input for the system during the training phase, while it is the control output when the system is operating. In this way, the network that controls the left hand will be switched off when low numbers 1—5 are processed. Note that the motor controller is implemented by two different RNNs, trained separately, but that we refer to as a single unit.
The use of RNNs to learn to count was investigated by Rodriguez et al. The artificial neural networks were implemented using the Matlab Neural Network Toolbox 8. The LMA is more robust than the GNA, which means that in many cases it finds a solution even if it starts far from the final minimum. Like the quasi-Newton methods, the LMA was designed to approach second-order training speed without having to compute the Hessian matrix.
In our implementation, the error e is calculated as the average of the squared errors of outputs. The Jacobian matrix can be computed through a standard backpropagation technique see Hagan and Menhaj, , that is much less complex than computing the Hessian matrix. Newton's method is faster and more accurate near an error minimum, so the aim is to shift toward Newton's method as quickly as possible.
In this way, the performance function is always reduced at each iteration of the algorithm. The derivative function of the RNN networks was the back-propagation through time Rumelhart et al. Naturally, it ensures all of the output values are between 0 and 1, and that their sum is 1. The softmax function used is a follows:. The architecture of the hidden layers of RNNs was chosen after a performance test, in which after runs with varying number of hidden neurons, the best trade-off solutions were selected in terms of minimization of the error and number of iterations needed to converge.
We found that 10 neurons was not surprisingly the ideal solution, this because 10 is also the number of different states to represent. Furthermore, in our preliminary experiments we also found that the pure linear transfer functions for the hidden layers were more effective than the usual sigmoid.
We chose not to use a bias or set them to zero for the RNN. Due to these choices, when the networks are not active, i. In addition to the main blocks, an associative network is included in the system to initiate the computation of the system and to implement the number manipulation. Indeed, after the RNNs have learned the number sequence, the switch is needed to stop the counting and to redirect the signals to the competitive classifier for the processing of the result. In particular, the unit is responsible for starting the counting by initializing all the hidden units to 1, and redirecting the hidden unit activation to the competitive classifier when the counting is finished.
Furthermore, this unit is crucial in the development for the acquisition of the ability to add numbers, because it can reset one of the two networks to make it count the new operand, and it lets the other continue as a buffer memory. These states will be studied in more detail in the number manipulation experiments. The table on the right summarizes the outputs according to the different states.
Bold lines indicate a full weighted connection, while normal lines are single connections. For simplicity hidden units of the two RNNs of the motor system are represented with one block. Vice versa if the initial state is set to 0, there is no activation because RNNs do not have bias in the hidden layer. A high value of the joint position represents the finger when it is closed, while low values indicate the finger is open.
Note that because of element collision and tendon limitations the actual values are not the ideal ones i. This is beneficial for the learning of weights and biases of the artificial neural networks. Using the material and methods presented above, first we studied the part of the cognitive system that learns to count.
For this first experiment, the main goal was to test the ability of the proposed cognitive system to learn numbers by comparing the performance of different ways of training the number knowledge of the robot with: 1 the internal representation hidden units activation of a given finger sequence, 2 the MFCC coefficients of number words out of sequence, 3 the internal representation of the number words sequence, 4 the internal representation of finger sequences plus the MFCC of number words out of sequence i.
To this end, we setup the experiment with the following steps: i the motor controller learns the opening of the fingers in a given sequence, in order to later establish a finger counting routine, and creates an internal representation for each step in the sequence by means of the hidden units activations; ii MFCCs are extracted from number words; iii the auditory memory learns the verbal number words in order from 1 to 10 and creates an internal representation for each word in the sequence.
From each learning step, relevant data are collected and stored as datasets for the experimentation, these sequences can be summarized as follows:. Internal representations of the words sequence: 10 values from hidden units' activations of auditory memory network. Internal motor representations of the finger sequence and MFCCs: a total of 23 values obtained by merging 1 and 2. Internal motor and auditory representations: a total of 20 inputs obtained by merging 1 and 3. Datasets 4 and 5 are built to model the learning when both fingers and number words are presented together as training input to the cognitive system.
Motor activations show a lateralization because the network that controls the left hand neurons 6—10 is switched off, furthermore the units from 1 to 5 remains fixed from the number five on because we suppose that the right hand is open we reason as if the robot is right handed. Meanwhile, the grouping of number words learned in or out of sequence is more random, and affects the learning as shown in the classification experiment. Hidden units' activation with the number sequence from 1 to A RNN trained with finger sequence. B RNN trained with word sequence. All datasets were used to train the competitive layer classifier to be classified in the ten classes that represent the numbers from 1 to The low number of epochs is, in this case, is imposed in order to study the robot's number learning in the early stages.
However, results show that 10 epochs are enough for the LMA to converge. However, if we consider as successful classification the cases when the class has the greatest likelihood, the only misclassification observed is for the number three. The t -test results confirm that all the differences are statistically significant except for the number three, when finger sequences are compared with word sequences, and the two when finger sequences only are compared with finger sequences and number words.
However, to learn number words in sequence helps to additionally improve the classification performance to highest likelihood, if internal representations are associated with motor ones. In order to study in more detail the development of learning, we measured the classification performance over the 10 epochs for the competitive layer trained with the different datasets. Looking at the developmental results, we once again see that number words learned out of a sequence are the less efficient to learn as there are no misclassifications only after 10 epochs, and the average likelihood is still low 0.
Conversely, if number words are learned in sequence and internal representations are used as inputs, the learning is faster in terms of precision of classification i. Indeed, the finger sequence reaches a higher average likelihood 0. Once the number sequences are learned, an interesting feature of the proposed cognitive system is the possibility to easily build up the ability to manipulate numbers with the development of the switch-associative network.
Indeed, this ability can be modeled by extending the capabilities of the associative network from the simple start and stop, to its transferring and mapping to the basic operation of addition. This can be considered, in a sense, an associative mapping between internal representations. This is implemented by activating a weighted connection between the two networks, which can be learned by applying the LMA to the two-layer network that comprises the hidden units of both networks. The operation of addition can be seen as a direct development of the concurrent learning of the two recurrent units motor and auditory.
Indeed, if one of the two does the actual counting of the operands, the other can be used as a buffer memory to add the result, when it is done, the final number can be transferred from the buffer to the other unit and then inputted to the final processor the classifier in our system. The first operand is heard by the auditory system and both networks will count until the corresponding activation of number 2 is reached. This step corresponds to the states of the associative network.
The sum operator is recognized so the auditory network is reset, while the first operand remains stored in the motor memory.
The second operand is heard, both networks restart to count as in step 1, until the auditory network reaches the activation corresponding to the number 2. In the meantime, the motor network reaches the activation of the number 4. After the auditory network stops, the associative network recognizes that the work is done so the total 4 is incepted from the fingers network to the auditory network thanks to the associative connection. Finally the output of the resulting number 4 is produced for final processing in our case the classifier.
The picture presents the steps done by the Robot's Cognitive System to perform the addition of two numbers, that in this example are both two.
The results obtained in our experiments with the iCub child-like robotic platform, show that learning the number words in sequence along with finger configurations helps the fast building of the initial representation of number in the robot. This does not mean that just learning the counting sequence from one to ten, is enough for children or our robot , to understand number concepts, but it is the repeated experience using the number word sequence when counting sets of things that is important in the development of numerical understanding Sarnecka and Carey, ; Donlan, While the use of fingers does not necessarily precede the use of language in the acquisition of a symbolic numerical system e.
Later on down the developmental path, with the child's early schooling experience, this mapping will also include written representations or numerals. These written representations, eventually take on the meaning of the spoken number word Fuson and Kwon, It is this kind of associative multi-modal learning that we are in a sense reproducing in our model. Studies focusing on how children acquire abstract words and concepts, have proposed that multiple representational systems involving both sensorimotor as well as linguistic information might be playing a role in conceptual representation e.
While the case of the acquisition of number words might be considered as a particular type of abstract word learning, theories such as the LASS theory Barsalou et al. In our model, number words or tags heard repeatedly, when coupled to the experience of moving the fingers, do serve as tools, used in the subsequent manipulation of the quantities they come to represent. In fact, the internal representations of the finger configurations themselves, found as a result of the experiments, can be considered to be a basis for the building of an embodied number representation in the robot, something in line with embodied and grounded cognition approaches to the study of mathematical cognitive processes.
Just as has been found with young children, through the use of finger counting and verbal counting strategies, our model develops finger and word representations that subsequently sustain the robot's learning the basic arithmetic operation of addition. Future work with the model will implement this.
This direction is stimulated by recent research on the simulation of mental imagery in cognitive systems and robots see Di Nuovo et al. The interested reader can find more details about this line of research in a recent special issue see Di Nuovo et al. In recent theoretical accounts of embodied numerosity e. The utility of children's learning finger counting strategies early in their mathematical education continues to be debated in mathematics education research, despite the evidence coming from neurocognitive and psychological studies indicating that it does for review of debate see Moeller et al.
Our experiments show that in fact, learning to count with the fingers, using verbal tags, can be helpful in the numerical training of a robot as well. While being inspired by the evidence from the studies we have cited in previous sections, our implementation is nonetheless, an abstraction of complex and as of yet not totally understood processes that may underlie the development of numerical cognition. Our results, however, are in line with what has been theoretically claimed in the developmental literature e. Examples of these principles and the role of finger counting that are relevant to our present study, and that we think we have simulated at least in part are: finger counting as an aid in the keeping track of the number words while reciting the counting sequence; as it contributing to the induction of the one-to-one correspondence principle by which children are helped by their fingers to coordinate the processes of tagging, or the attribution of a number word to each item; and as facilitating the assimilation of the stable-order principle where numerical labels have to be enumerated in the same order across counting sequences see also Andres et al.
The study of mathematical cognitive processes, traditionally considered to be quintessential examples of abstract and symbolic processing, have been assumed to primarily involve the mind rather than the body. Our embodied robot experiments indicate, that aspects of the development of this knowledge can be accounted for not only by way of bodily representations, but also with an artificial network in the place of a mind.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. This work was supported in part by the European Commission under Grants n. National Center for Biotechnology Information , U. Journal List Front Behav Neurosci v. Front Behav Neurosci.
Published online Feb 3. Prepublished online Nov Vivian M.
- Making fingers and words count in a cognitive robot.
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De La Cruz. Author information Article notes Copyright and License information Disclaimer. Reviewed by: Anna M. Received Sep 30; Accepted Jan 8. The use, distribution or reproduction in other forums is permitted, provided the original author s or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
This article has been cited by other articles in PMC. Abstract Evidence from developmental as well as neuroscientific studies suggest that finger counting activity plays an important role in the acquisition of numerical skills in children. Keywords: embodied cognition, developmental robotics, finger counting, number words, number cognition. Introduction Whether finger counting is an essential stage in the development of the cognition of number is still highly debated, though strong evidence exists on the positive contribution of sensorimotor skills and representation in numerical cognition.
Materials and methods The robotic model used for the experiments is a computer simulation model of the iCub humanoid robot Tikhanoff et al. Open in a separate window. Figure 1. Figure 2. Figure 3. Table 1 Actual finger joint positions according to number representation.
Table 2 Mel Frequency Cepstral Coefficients for number words. Experiments and results Using the material and methods presented above, first we studied the part of the cognitive system that learns to count. Numbers learning For this first experiment, the main goal was to test the ability of the proposed cognitive system to learn numbers by comparing the performance of different ways of training the number knowledge of the robot with: 1 the internal representation hidden units activation of a given finger sequence, 2 the MFCC coefficients of number words out of sequence, 3 the internal representation of the number words sequence, 4 the internal representation of finger sequences plus the MFCC of number words out of sequence i.
MFCCs from number words: 13 values, not as part of a sequence. Figure 4. Figure 5. Optimal leaf-order of hidden units' activations. Table 3 Likelihood of number classes with different training datasets. Figure 6. Average likelihood with number classes with varying epochs. Figure 7. Median number misclassifications with varying epochs of learning.
Numbers manipulation Once the number sequences are learned, an interesting feature of the proposed cognitive system is the possibility to easily build up the ability to manipulate numbers with the development of the switch-associative network. Figure 8. Discussion The results obtained in our experiments with the iCub child-like robotic platform, show that learning the number words in sequence along with finger configurations helps the fast building of the initial representation of number in the robot.